11

THE MODELLING POTENTIAL THE MODELLING POTENTIAL OF MINIMAX ALGEBRAOF MINIMAX ALGEBRA

H.P. WilliamsH.P. Williams

London School of EconomicsLondon School of Economics

22

MaxMax

MinMin

++

-- // //

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

33

Connections between Minimax Algebra and Connections between Minimax Algebra and Mathematical ProgrammingMathematical Programming

1. Mathematical Programming is concerned with models of form

)""unlike(i.e.

dbca

dbca

dc

ba

Duality Theorem of LP demonstrates inequalities can be added in multiples to give tight bound on optimal objective value

ixgxf

i 0)(:tosubject)(Max/Min

2. Can add but not subtract inequalities

44

The Dual of a Disjunctive Programme (in The Dual of a Disjunctive Programme (in Disjunctive Normal FormDisjunctive Normal Form))

0j

jj

x

bxa

xc

kijkijik

j

Minimise

Subject to:

Dual is

Maximise kii

kik

ybSubject to:

0ki

k ikij

y

ca

If Primal and Dual Solvable (not Infeasible or Unbounded) they have the same optimal objective value.

ik ,

j

ik ,

'

j j

55

Dynamic ProgrammingDynamic ProgrammingSpecial cases lend themselves to Mini Max formulations

Knapsack Problem

11

1

1

,1,Max),1(

:Subject to

Maximise),(

nnn

j

n

jj

j

n

jj

cabnfbfbnf

bxa

xcbnf

(a nested recursion)

Can be written as

),(//,1,1 11 bnfabnfcbnf nn

66

The Group Knapsack ProblemThe 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. gxgojj

n

ij

Where are members of a finite abelian group gj

77

The Group Knapsack ProblemThe 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

jx

j

n

j

jxc1

jc

As with conventional Knapsack Problem can be solved by Dynamic Programming

)()(( ,//,11 1),1 bnagnn ffnf ncg

88

The Group Knapsack ProblemThe Group Knapsack Problem

5. But resultant solution may imply negative (infeasible) values for basic variables.

Need therefore to seek 2nd best, 3rd 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.

99

Shortest Path ProblemShortest Path ProblemAll deterministic Dynamic Programmes (including Knapsack problems) can be formulated as Shortest Path problems

0

.

.

..

0

)( arc oflength is

(t) node terminalto node frompath shortest is

dd

ei

if

ifdaMindijd

id

jij

j

i

i

i

Different Linear Algebra methods mirror different Shortest Path methods

trow

tj tj

1010

Minimax ProblemsMinimax Problems

Facility Location, Obnoxious Facilities, Political Districting, Nucleolus of a Game, ‘Fair’ Allocations.

ii

i xMinxMaxx

Min

Analogous to conventional objectives

1111

LL1 1 and L Normsand L Norms

Can formulate Linear Regression problems using Minimax Algebra

Special Case: Zero Dimension problems

Median of a Set of Numbers

)(,...,, 121 NormLyyy n

Pi Pi

Pii yyMaxNNorm is a partition of numbers into sets of

PP :

)(2

1

)(2

oddnnumbersn

or

evennnumbersn

i.e. Maximum ‘Contrast’

Pii

Pi

i

y

yN

1212

L NormL Norm

j

i

ji

jiji

y

yN

yyMaxNNorm

,

,2

1

1313

Scheduling ProblemsScheduling Problems

Jobs cannot start (e.g. Until previous jobs (e.g. ) have finished (durations )

3x 21, xx21,dd

22113

22113

..,

xdxdxeidxdxMaxx

May wish to Minimise completion of all jobs

iii

i

xdMinei

xdMaxMinge

..

.. 11

Mirrors an LP model

x

1414

ReferencesReferences

BA CarrBA Carré, An Algebra for Network é, An Algebra for Network Routing Problems, Routing Problems, J Inst Math Appl J Inst Math Appl 7 7 (1971) 273-293(1971) 273-293

RA Cunningham Green RA Cunningham Green Minimax Algebra Minimax Algebra Vol. 166. Lecture Notes in Economics and Vol. 166. Lecture Notes in Economics and Mathematical Systems, Springer Verlag Mathematical Systems, Springer Verlag 19791979