1. 1 THE MODELLING POTENTIAL OF MINIMAX ALGEBRA H.P. Williams London School of Economics

2. 2 MaxMin + - // - // N.B.(unlike +) does not have an inverse. Hence we are concerned with semi rings.

3. 3 Connections between Minimax Algebra and Mathematical Programming 1.Mathematical Programming is concerned with models of form Duality Theorem of LP demonstrates inequalities can be added in multiples to give tight bound on optimal objective value 2.Can add but not subtract inequalities

4. 4 The Dual of a Disjunctive Programme (in Disjunctive Normal Form) Minimise Subject to: Dual is Maximise Subject to: If Primal and Dual Solvable (not Infeasible or Unbounded) they have the same optimal objective value.

5. 5 Dynamic Programming Special cases lend themselves to Mini Max formulations Knapsack Problem (a nested recursion) Can be written as

6. 6 The Group Knapsack Problem 1.Solve LP Relaxation of an Integer Programme 2.Choose integral, non-negative values for non-basic variables so as to make basic variables integral 3.This gives a set of congruence relations i.e. a group equation (in non-negative variables) i.e. Where are members of a finite abelian group

7. 7 The Group Knapsack Problem 4.If we choose so as to minimise difference from optimal LP objective we have an objective Minimise Where are LP reduced costs of non-basic variables As with conventional Knapsack Problem can be solved by Dynamic Programming

8. 8 The Group Knapsack Problem 5. But resultant solution may imply negative (infeasible) values for basic variables. Need therefore to seek 2 nd best, 3 rd best etc. solution to Group Knapsack Problem 6.Cuninghame-Green does this systematically (‘Integer Programming by Long Division’) by successively enumerating solutions of monotonically increasing cost until a feasible solutions if found.

9. 9 Shortest Path Problem All deterministic Dynamic Programmes (including Knapsack problems) can be formulated as Shortest Path problems Different Linear Algebra methods mirror different Shortest Path methods

10. 10 Minimax Problems Facility Location, Obnoxious Facilities, Political Districting, Nucleolus of a Game, ‘Fair’ Allocations. Analogous to conventional objectives

11. 11 L 1 and L Norms Can formulate Linear Regression problems using Minimax Algebra Special Case: Zero Dimension problems Median of a Set of Numbers is a partition of numbers into sets of i.e. Maximum ‘Contrast’

12. 12 L Norm

13. 13 Scheduling Problems Jobs cannot start (e.g. Until previous jobs (e.g. ) have finished (durations ) May wish to Minimise completion of all jobs Mirrors an LP model

14. 14 References  BA Carré, An Algebra for Network Routing Problems, J Inst Math Appl 7 (1971) 273-293  RA Cunningham Green Minimax Algebra Vol. 166. Lecture Notes in Economics and Mathematical Systems, Springer Verlag 1979