8 research outputs found
The integer knapsack cover polyhedron
We study the integer knapsack cover polyhedron which is the convex hull of the set of vectors x ∈ ℤ+ n that satisfy C T x ≥ b, with C ∈ ℤ++ n and 6 ∈ ℤ++. We present some general results about the nontrivial facet-defining inequalities. Then we derive specific families of valid inequalities, namely, rounding, residual capacity, and lifted rounding inequalities, and identify cases where they define facets. We also study some known families of valid inequalities called 2-partition inequalities and improve them using sequence-independent lifting. © 2007 Society for Industrial and Applied Mathematics
Manufacturer's mixed pallet design problem
Cataloged from PDF version of article.We study a problem faced by a major beverage producer. The company produces and distributes several brands to various
customers from its regional distributors. For some of these brands, most customers do not have enough demand to
justify full pallet shipments. Therefore, the company decided to design a number of mixed or ‘‘rainbow’’ pallets so that its
customers can order these unpopular brands without deviating too much from what they initially need. We formally state
the company’s problem as determining the contents of a pre-determined number of mixed pallets so as to minimize the
total inventory holding and backlogging costs of its customers over a finite horizon. We first show that the problem is
NP-hard. We then formulate the problem as a mixed integer linear program, and incorporate valid inequalities to
strengthen the formulation. Finally, we use company data to conduct a computational study to investigate the efficiency
of the formulation and the impact of mixed pallets on customers’ total costs.
2007 Elsevier B.V. All rights reserved
The Robust Network Loading Problem under Hose Demand Uncertainty: Formulation, Polyhedral Analysis, and Computations
Cataloged from PDF version of article.We consider the network loading problem (NLP) under a polyhedral uncertainty description of traffic
demands. After giving a compact multicommodity flow formulation of the problem, we state a decomposition
property obtained from projecting out the flow variables. This property considerably simplifies the
resulting polyhedral analysis and computations by doing away with metric inequalities. Then we focus on a
specific choice of the uncertainty description, called the “hose model,” which specifies aggregate traffic upper
bounds for selected endpoints of the network. We study the polyhedral aspects of the NLP under hose demand
uncertainty and use the results as the basis of an efficient branch-and-cut algorithm. The results of extensive
computational experiments on well-known network design instances are reported
The robust network loading problem under hose demand uncertainty: Formulation, polyhedral analysis, and computations
We consider the network loading problem (NLP) under a polyhedral uncertainty description of traffic demands. After giving a compact multicommodity flow formulation of the problem, we state a decomposition property obtained from projecting out the flow variables. This property considerably simplifies the resulting polyhedral analysis and computations by doing away with metric inequalities. Then we focus on a specific choice of the uncertainty description, called the "hose model," which specifies aggregate traffic upper bounds for selected endpoints of the network. We study the polyhedral aspects of the NLP under hose demand uncertainty and use the results as the basis of an efficient branch-and-cut algorithm. The results of extensive computational experiments on well-known network design instances are reported. © 2011 INFORMS
Formulations and valid inequalities for the heterogeneous vehicle routing problem
We consider the vehicle routing problem where one can choose among vehicles with different costs and capacities to serve the trips. We develop six different formulations: the first four based on Miller-Tucker-Zemlin constraints and the last two based on flows. We compare the linear programming bounds of these formulations. We derive valid inequalities and lift some of the constraints to improve the lower bounds. We generalize and strengthen subtour elimination and generalized large multistar inequalities
Robust network design under polyhedral traffic uncertainty
Ankara : The Department of Industrial Engineering and The Institute of Engineering and Science of Bilkent Univ., 2007.Thesis (Ph.D.) -- Bilkent University, 2007.Includes bibliographical references leaves 160-166.In this thesis, we study the design of networks robust to changes in demand
estimates. We consider the case where the set of feasible demands is defined by
an arbitrary polyhedron. Our motivation is to determine link capacity or routing
configurations, which remain feasible for any realization in the corresponding
demand polyhedron. We consider three well-known problems under polyhedral
demand uncertainty all of which are posed as semi-infinite mixed integer programming
problems. We develop explicit, compact formulations for all three problems
as well as alternative formulations and exact solution methods.
The first problem arises in the Virtual Private Network (VPN) design field.
We present compact linear mixed-integer programming formulations for the problem
with the classical hose traffic model and for a new, less conservative, robust
variant relying on accessible traffic statistics. Although we can solve these formulations
for medium-to-large instances in reasonable times using off-the-shelf MIP
solvers, we develop a combined branch-and-price and cutting plane algorithm to
handle larger instances. We also provide an extensive discussion of our numerical
results.
Next, we study the Open Shortest Path First (OSPF) routing enhanced with
traffic engineering tools under general demand uncertainty with the motivation to
discuss if OSPF could be made comparable to the general unconstrained routing
(MPLS) when it is provided with a less restrictive operating environment. To
the best of our knowledge, these two routing mechanisms are compared for the
first time under such a general setting. We provide compact formulations for
both routing types and show that MPLS routing for polyhedral demands can
be computed in polynomial time. Moreover, we present a specialized branchand-price
algorithm strengthened with the inclusion of cuts as an exact solution tool. Subsequently, we compare the new and more flexible OSPF routing with
MPLS as well as the traditional OSPF on several network instances. We observe
that the management tools we use in OSPF make it significantly better than the
generic OSPF. Moreover, we show that OSPF performance can get closer to that
of MPLS in some cases.
Finally, we consider the Network Loading Problem (NLP) under a polyhedral
uncertainty description of traffic demands. After giving a compact multicommodity
formulation of the problem, we prove an unexpected decomposition
property obtained from projecting out the flow variables, considerably simplifying
the resulting polyhedral analysis and computations by doing away with metric inequalities,
an attendant feature of most successful algorithms on NLP. Under the
hose model of feasible demands, we study the polyhedral aspects of NLP, used as
the basis of an efficient branch-and-cut algorithm supported by a simple heuristic
for generating upper bounds. We provide the results of extensive computational
experiments on well-known network design instances.Altın, AyşegülPh.D