Browsing by Subject "Min-max"

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Open Access
The multi-terminal maximum-flow network-interdiction problem
(2011) Akgün, İ.; Tansel, B. Ç.; Wood, R. K.
This paper defines and studies the multi-terminal maximum-flow network-interdiction problem (MTNIP) in which a network user attempts to maximize flow in a network among K ≥ 3 pre-specified node groups while an interdictor uses limited resources to interdict network arcs to minimize this maximum flow. The paper proposes an exact (MTNIP-E) and an approximating model (MPNIM) to solve this NP-hard problem and presents computational results to compare the models. MTNIP-E is obtained by first formulating MTNIP as bi-level min-max program and then converting it into a mixed integer program where the flow is explicitly minimized. MPNIM is binary-integer program that does not minimize the flow directly. It partitions the node set into disjoint subsets such that each node group is in a different subset and minimizes the sum of the arc capacities crossing between different subsets. Computational results show that MPNIM can solve all instances in a few seconds while MTNIP-E cannot solve about one third of the problems in 24 hour. The optimal objective function values of both models are equal to each other for some problems while they differ from each other as much as 46.2% in the worst case. However, when the post-interdiction flow capacity incurred by the solution of MPNIM is computed and compared to the objective value of MTNIP-E, the largest difference is only 7.90% implying that MPNIM may be a very good approximation to MTNIP-E. © 2011 Elsevier B.V. All rights reserved.
Open Access
Optimization of transportation requirements in the deployment of military units
(2005) Akgün, İbrahim
We study the deployment planning problem (DPP) that may roughly be defined as the problem of the planning of the physical movement of military units, stationed at geographically dispersed locations, from their home bases to their designated destinations while obeying constraints on scheduling and routing issues as well as on the availability and use of various types of transportation assets that operate on a multimodal transportation network. The DPP is a large-scale real-world problem for which no analytical models are existent. In this study, we define the problem in detail and analyze it with respect to the academic literature. We propose three mixed integer programming models with the objectives of cost, lateness (the difference between the arrival time of a unit and its earliest allowable arrival time at its destination), and tardiness (the difference between the arrival time of a unit and its latest arrival time at its destination) minimization to solve the problem. The cost-minimization model minimizes total transportation cost of a deployment and is of use for investment decisions in transportation resources during peacetime and for deployment planning in cases where the operation is not imminent and there is enough time to do deliberate planning that takes costs into account. The lateness and tardiness minimization models are of min-max type and are of use when quick deployment is of utmost concern. The lateness minimization model is for cases when the given fleet of transportation assets is sufficient to deploy units within their allowable time windows and the tardiness minimization model is for cases when the given fleet is not sufficient. We propose a solution methodology for solving all three models. The solution methodology involves an effective use of relaxation and restriction that significantly speeds up a CPLEX-based branchand-bound. The solution times for intermediate sized problems are around one hour at maximum for cost and lateness minimization models and around two hours for the tardiness minimization model. Producing a suboptimal feasible solution based on trial and error methods for a problem of the same size takes about a week in the current practice in the Turkish Armed Forces. We also propose a heuristic that is essentially based on solving the models incrementally rather than at one step. Computational results show that the heuristic can be used to find good feasible solutions for the models. We conclude the study with comments on how to use the models in the realworld.