Igor Averbakh

Interval data minmax regret network optimization problems

We consider the minimum spanning tree and the shortest path problems on a network with uncertain lengths of edges. In particular, for any edge of the network, only an interval estimate of the length of the edge is known, and it is assumed that the length of each edge can take on any value from the corresponding interval of uncertainty, regardless of the values taken by the lengths of other edges. It is required to find a minmax regret solution. We prove that both problems are NP-hard even if the bounds of all intervals of uncertainty belong to {0,1}. The interval data minmax regret shortest path problem is NP-hard even if the network is directed, acyclic, and has a layered structure. We show that the problems are polynomially solvable in the practically important case where the number of edges with uncertain lengths is fixed or is bounded by the logarithm of a polynomial function of the total number of edges. We discuss implications of these results for the general theory of interval data minmax regret combinatorial optimization.

Minmax regret solutions for minimax optimization problems with uncertainty

We propose a general approach for finding minmax regret solutions for a class of combinatorial optimization problems with an objective function of minimax type and uncertain objective function coefficients. The approach is based on reducing a problem with uncertainty to a number of problems without uncertainty. The method is illustrated on bottleneck combinatorial optimization problems, minimax multifacility location problems and maximum weighted tardiness scheduling problems with uncertainty.

Minimax regret p-center location on a network with demand uncertainty

Abstract We consider the weighted p-center problem on a transportation network with uncertain weights of nodes. Specifically, for each node, an interval estimate of its weight is known. The objective is to find the ‘minimax regret’ solution i.e. to minimize the worst-case loss in the objective function that may occur because a decision is made without knowing which state of nature will take place. We discuss properties of the problem and show that the problem can be solved by means of solving (n + 1) regular weighted p-center problems. This leads to polynomial algorithms for the cases where the regular weighted p-center problem can be solved in polynomial time, e.g. for the case of a tree network, and for the case of a general network with p = 1.

Algorithms for the robust 1-center problem on a tree

Abstract We consider the weighted 1-center problem on a network with uncertainty in node weights and edge lengths. Uncertainty is modelled by means of interval estimates for parameters. Specifically, each uncertain parameter is assumed to be random with unknown distribution and can take on any value within a corresponding prespecified interval. It is required to find a robust (minmax regret) solution, that is, a location which is e -optimal for any possible realization of parameters, with e as small as possible. The problem on a general network is known to be NP-hard; for the problem on a tree, we present a polynomial algorithm.

A heuristic with worst-case analysis for minimax routing of two travelling salesmen on a tree

Abstract Suppose two travelling salesmen must visit together all points/nodes of a tree, and the objective is to minimize the maximal length of their tours. Home locations of the salesmen are given, and it is required to find optimal tours. For this NP-hard problem a heuristic with complexity O(n) is presented. The worst-case relative error for the heuristic performance is 1 3 for the case of equal home locations for both servers and 1 2 for the case of different home locations.

( p -1)/( p +1)-approximate algorithms for p -traveling salemen problems on a tree with minmax objective

Abstract Suppose p traveling salesmen must visit together all points/nodes of a tree, and the objective is to minimize the maximum of lengths of their tours. For location-allocation problems (where both optimal home locations of the salesmen and their tours must be found), which are NP-complete, fast polynomial heuristics with worst-case relative error (p − 1) (p + 1) are presented.

On-line supply chain scheduling problems with preemption

We consider supply chain scheduling problems where customers release jobs to a manufacturer that has to process the jobs and deliver them to the customers. The jobs are released on-line, that is, at any time there is no information on the number, release and processing times of future jobs; the processing time of a job becomes known when the job is released. Preemption is allowed. To reduce the total costs, processed jobs are grouped into batches, which are delivered to customers as single shipments; we assume that the cost of delivering a batch does not depend on the number of jobs in the batch. The objective is to minimize the total cost, which is the sum of the total flow time and the total delivery cost. For the single-customer problem, we present an on-line two-competitive algorithm, and show that no other on-line algorithm can have a better competitive ratio. We also consider an extension of the algorithm for the case of m customers, and show that its competitive ratio is not greater than 2m if the delivery costs to different customers are equal.

On the complexity of minmax regret linear programming

Abstract We consider linear programming problems with uncertain objective function coefficients. For each coefficient of the objective function, an interval of uncertainty is known, and it is assumed that any coefficient can take on any value from the corresponding interval of uncertainty, regardless of the values taken by other coefficients. It is required to find a minmax regret solution. This problem received considerable attention in the recent literature, but its computational complexity status remained unknown. We prove that the problem is strongly NP-hard. This gives the first known example of a minmax regret optimization problem that is NP-hard in the case of interval-data representation of uncertainty but is polynomially solvable in the case of discrete-scenario representation of uncertainty.

Complexity of minimizing the total flow time with interval data and minmax regret criterion

We consider the minmax regret (robust) version of the problem of scheduling n jobs on a machine to minimize the total flow time, where the processing times of the jobs are uncertain and can take on any values from the corresponding intervals of uncertainty. We prove that the problem in NP-hard. For the case where all intervals of uncertainty have the same center, we show that the problem can be solved in O(n log n) time if the number of jobs is even, and is NP-hard if the number of jobs is odd. We study structural properties of the problem and discuss some polynomially solvable cases.

On-line integrated production-distribution scheduling problems with capacitated deliveries

In on-line integrated production-distribution problems, customers release jobs to a manufacturer that has to process the jobs and deliver them to the customers. The jobs are released on-line, that is, at any time there is no information about future jobs. Processed jobs are grouped into batches, which are delivered to the customers as single shipments. The total cost (to be minimized) is the sum of the total weighted flow time and the total delivery cost. Such on-line integrated production-distribution problems have been studied for the case of uncapacitated batches. We consider the capacitated case with an upper bound on the size of a batch. For several versions of the problem, we present efficient on-line algorithms, and use competitive analysis to study their worst-case performance.