Optimal Sequenced Matroid Bases Solved by Ga’s with Feasible Search Space including Applications

We consider an extension to the optimal matroid base problem [1] whereby the matroid element costs are not fixed, but are time dependent. We propose a genetic algorithm (GA) approach to solve the optimal sequenced matroid base problem (OSMBP) by employing efficient codes which are suffixed by a standard permutation code [2]. These novel encoding schemes insure feasibility after performing the classical operations of crossover and mutation and also ensure the feasibility of the initial randomly generated population (i.e., generation 0). This class of problems, where costs are not fixed but are time dependent, embrace non-locality which actually makes the GAs more efficient. A variety of practical matroid applications with time dependent costs will also be presented. Introduction A finite matroid M is a pair (E,β) where E is a nonempty finite set and β is a non-empty collection of subsets of E called bases satisfying the following properties: i) no base properly contains another base, and ii) if B1 and B2 are bases in β (i.e., B1, B2 ∈ β) and if e∈B1 then there exists f ∈B2 such that ((B1 – {e}) ∪ {f}) ∈ β (i.e., ((B1 – {e}) ∪ {f}) is also a base). By repeatedly using ii) it is easily shown that any two bases B1, B2 ∈ β have the same number of elements. This number is called the rank of M. Any subset of a base is called an independent set. The dual matroid M = (E,β) of M over E has its bases in β = { B B ∈ β }, that is, the collection of the complements of β [1]. Given M = (E,β), the optimal matroid base problem (OMBP) is to find an optimal base (not necessarily unique)of M (i.e., a base B with a maximu (minimum) total cost (or weight)) where each element of e∈E has a cost (or weight) which is a real number C(e). Kruskal’s algorithm applied to matroids solves this problem as well as simultaneously solving the OMBP for M (which has an optimal base (B ) with a minimum (maximum) total cost (or weight)). Here we consider an extension to the optimal matroid base problem (OMBP) whereby the matroid element costs are not fixed, but are time dependent. This is called the optimal sequenced matroid base problem (OSMBP). Let M = (E,β) be a matroid of rank n such that each element e∈E has assigned to it C(e,t) a real valued function of t having domain { 1, 2, ..., n }. We refer to M as a matroid with time dependent cost assignment C and we say C(e,t) is the cost of selecting e at time t. A sequencing (permutation, ordering) of the elements of a base B∈β is called a sequenced (ordered) base. It is assumed that the selection of any e∈E takes exactly one unit of processing time. A sequenced base is an n-tuple (e1, e2, ..., en) which shows the order in which the elements of the base are to be selected and determines the costs C(e1,1), C(e2,2), ..., C(en,n) associated with this sequence. The total cost C(e1,1) + C(e2,2) + ...+ C(en,n) is the objective function T(e1, e2,..., en) to be optimized. Since this problem is NP hard, we propose a genetic algorithm (GA) to solve the optimal sequence matroid base problem (OSMBP) using efficient codes which are suffixed by a permutation code [2]. These novel encoding schemes insure feasibility after performing the classical operations of crossover and mutation and also ensure the feasibility of the initial randomly generated population. These types of problems, where costs are not fixed but are time dependent, embrace the non-locality which is implied in our coding and which makes the GAs more efficient. A wide variety of pragmatic matroid applications with time dependent costs will be presented (i.e., spanning tree, airline route sales (dual matroid), position assignment, node base communications, and bidding with slack time applications). (Note: A similar problem of less complexity has been solved in polynomial time in [ 3]). References 1. Wilson, R.J. ,Graph Theory, Fourth Ed, Addison Wesley , 2000 2. Edelson, W. , and Gargano, M.L. , Minimal EdgeOrdered Spanning Trees Solved By a Genetic Algorithm With Feasible Search Space, Congressus Numerantium, 135 (1998) pp 37-45. 3.Gargano, M.L., Quintas, L.V. , and Friedrich, S.C., Matroid Bases With Optimal Sequencing, Congressus Numerantium 82, (1991) pp 65-77. 1447 REAL WORLD APPLICATIONS: POSTER PAPERS