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1.
In this paper we consider the familiar bin-packing problem and its associated set-partitioning formulation. We show that the
optimal solution to the bin-packing problem can be no larger than 4/3 ⌈Z
LP⌉, whereZ
LP is the optimal solution value of the linear programming relaxation of the set-partitioning formulation. An example is provided
to show that the bound is tight. A by-product of our analysis is a new worst-case bound on the performance of the well studied
First Fit Decreasing and Best Fit Decreasing heuristics.
This research was supported in part by ONR Contracts N00014-90-J-1649 and N00014-95-1-0232, and NSF Contracts DDM-8922712
and DDM-9322828. 相似文献
2.
Deniz Aksen Nuray Piyade Necati Aras 《Central European Journal of Operations Research》2010,18(3):269-291
In this article, we elaborate on a budget constrained extension of the r-interdiction median problem with fortification (RIMF). The objective in the RIMF is to find the optimal allocation of protection
resources to a given service system consisting of p facilities so that the disruptive effects of r possible attacks to the system are minimized. The defender of the system needs to fortify q facilities of the present system to offset the worst-case loss of r non-fortified facilities due to an interdiction in which the attacker’s objective is to cause the maximum possible disruption
in the service level of the system. The defender-attacker relationship fits a bilevel integer programming (BIP) formulation
where the defender and attacker take on the respective roles of the leader and the follower. We adopt this BIP formulation
and augment it with a budget constraint instead of a predetermined number of facilities to be fortified. In addition, we also
assume that each facility has a flexible service capacity, which can be expanded at a unit cost to accommodate the demand
of customers who were serviced by some other interdicted facility before the attack. First, we provide a discrete optimization
model for this new facility protection planning scenario with a novel set of closest assignment constraints. Then, to tackle
this BIP problem we use an implicit enumeration algorithm performed on a binary tree. For each node representing a different
fortification scheme, the attacker’s problem is solved to optimality using Cplex 11. We report computational results obtained
on a test bed of 96 randomly generated instances. The article concludes with suggestions for future research. 相似文献
3.
In this paper, we consider the capacitated multi-facility Weber problem with rectilinear distance. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the rectilinear distance separating them. We first give a new mixed integer linear programming formulation of the problem by making use of a well-known necessary condition for the optimal facility locations. We then propose new heuristic solution methods based on this formulation. Computational results on benchmark instances indicate that the new methods can provide very good solutions within a reasonable amount of computation time. 相似文献
4.
Optimal location with equitable loads 总被引:1,自引:0,他引:1
Oded Berman Zvi Drezner Arie Tamir George O. Wesolowsky 《Annals of Operations Research》2009,167(1):307-325
The problem considered in this paper is to find p locations for p facilities such that the weights attracted to each facility will be as close as possible to one another. We model this problem
as minimizing the maximum among all the total weights attracted to the various facilities. We propose solution procedures
for the problem on a network, and for the special cases of the problem on a tree or on a path. The complexity of the problem
is analyzed, O(n) algorithms and an O(pn
3) dynamic programming algorithm are proposed for the problem on a path respectively for p=2 and p>2 facilities. Heuristic algorithms (two types of a steepest descent approach and tabu search) are proposed for its solution.
Extensive computational results are presented. 相似文献
5.
In the p-center problem, it is assumed that the facility located at a node responds to demands originating from the node. This assumption
is suitable for emergency and health care services. However, it is not valid for large-scale emergencies where most of facilities
in a whole city may become functionless. Consequently, residents in some areas cannot rely on their nearest facilities. These
observations lead to the development of a variation of the p-center problem with an additional assumption that the facility at a node fails to respond to demands from the node. We use
dynamic programming approach for the location on a path network and further develop an efficient algorithm for optimal locations
on a general network. 相似文献
6.
A well known formulation of the multiple sequence alignment (MSA) problem is the maximum weight trace (MWT), a 0–1 linear
programming problem. In this paper, we propose a new integer quadratic programming formulation of the MSA. The number of constraints
and variables in the problem are only of the order of kL
2, where, k is the number of sequences and L is the total length of the sequences, that is, L = ?i=1kli{L= \sum_{i=1}^{k}l_{i}} , where l
i
is the length of sequence i. Based on this formulation we introduce an equivalent linear constrained 0–1 quadratic programming problem. We also propose
a 0–1 linear programming formulation of the MWT problem, with polynomially many constraints. Our formulation provides the
first direct compact formulation that ensures that the critical circuit inequalities (which are exponentially many) are all
met. 相似文献
7.
We consider the maximization of a multicommodity flow throughput in presence of constraints on the maximum number of paths
to be used. Such an optimization problem is strongly NP-hard, and is known in the literature as the maximum routable demand fraction variant of the k-splittable flow problem. Here we propose an exact approach based on branch and bound rules and on an arc-flow mixed integer
programming formulation of the problem. Computational results are provided, and a comparison with a standard commercial solver
is proposed. 相似文献
8.
An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts 总被引:6,自引:0,他引:6
This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning
formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding
procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining
three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing
and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing
and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose
reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved
by an integer programming solver. Computational results over the main instances from the literature show the effectiveness
of the proposed algorithm.
相似文献
9.
Mark-Christoph Körner Juan A. Mesa Federico Perea Anita Schöbel Daniel Scholz 《TOP》2014,22(1):227-253
In this paper the following facility location problem in a mixed planar-network space is considered: We assume that traveling along a given network is faster than traveling within the plane according to the Euclidean distance. A pair of points (A i ,A j ) is called covered if the time to access the network from A i plus the time for traveling along the network plus the time for reaching A j is lower than, or equal to, a given acceptance level related to the travel time without using the network. The objective is to find facilities (i.e. entry and exit points) on the network that maximize the number of covered pairs. We present a reformulation of the problem using convex covering sets and use this formulation to derive a finite dominating set and an algorithm for locating two facilities on a tree network. Moreover, we adapt a geometric branch and bound approach to the discrete nature of the problem and suggest a procedure for locating more than two facilities on a single line, which is evaluated numerically. 相似文献
10.
Alper Atamtürk 《Mathematical Programming》2006,108(2-3):235-250
We introduce strong formulations for robust mixed 0–1 programming with uncertain objective coefficients. We focus on a polytopic
uncertainty set described by a ``budget constraint' for allowed uncertainty in the objective coefficients. We show that for
a robust 0–1 problem, there is an α–tight linear programming formulation with size polynomial in the size of an α–tight linear programming formulation for the nominal 0–1 problem. We give extensions to robust mixed 0–1 programming and
present computational experiments with the proposed formulations. 相似文献
11.
《European Journal of Operational Research》2006,168(3):716-731
The simple assembly line balancing problem is a classical integer programming problem in operations research. A set of tasks, each one being an indivisible amount of work requiring a number of time units, must be assigned to workstations without exceeding the cycle time. We present a new lower bound, namely the LP relaxation of an integer programming formulation based on Dantzig–Wolfe decomposition. We propose a column generation algorithm to solve the formulation. Therefore, we develop a branch-and-bound algorithm to exactly solve the pricing problem. We assess the quality of the lower bound by comparing it with other lower bounds and the best-known solution of the various instances from the literature. Computational results show that the lower bound is equal to the best-known objective function value for the majority of the instances. Moreover, the new LP based lower bound is able to prove optimality for an open problem. 相似文献
12.
We examine a routing problem in which network arcs fail according to independent failure probabilities. The reliable h-path routing problem seeks to find a minimum-cost set of h ≥ 2 arc-independent paths from a common origin to a common destination, such that the probability that at least one path
remains operational is sufficiently large. For the formulation in which variables are used to represent the amount of flow
on each arc, the reliability constraint induces a nonconvex feasible region, even when the integer variable restrictions are
relaxed. Prior arc-based models and algorithms tailored for the case in which h = 2 do not extend well to the general h-path problem. Thus, we propose two alternative integer programming formulations for the h-path problem in which the variables correspond to origin-destination paths. Accordingly, we develop two branch-and-price-and-cut
algorithms for solving these new formulations, and provide computational results to demonstrate the efficiency of these algorithms. 相似文献
13.
We consider the network design problem which consists in determining at minimum cost a 2-edge connected network such that
the shortest cycle (a “ring”) to which each edge belongs, does not exceed a given length K. We identify a class of inequalities, called cycle inequalities, valid for the problem and show that these inequalities together
with the so-called cut inequalities yield an integer programming formulation of the problem in the space of the natural design
variables. We then study the polytope associated with that problem and describe further classes of valid inequalities. We
give necessary and sufficient conditions for these inequalities to be facet defining. We study the separation problem associated
with these inequalities. In particular, we show that the cycle inequalities can be separated in polynomial time when K≤4. We develop a Branch-and-Cut algorithm based on these results and present extensive computational results. 相似文献
14.
The standard quadratic program (QPS) is
minxεΔxTQx, where
is the simplex Δ = {x ⩽ 0 ∣ ∑i=1n xi = 1}. QPS can be used to formulate combinatorial problems such as the maximum stable set problem, and also arises in global
optimization algorithms for general quadratic programming when the search space is partitioned using simplices. One class
of ‘d.c.’ (for ‘difference between convex’) bounds for QPS is based on writing Q=S−T, where S and T are both positive semidefinite, and bounding
xT Sx (convex on Δ) and −xTx
(concave on Δ) separately. We show that the maximum possible such bound can be obtained by solving a semidefinite programming
(SDP) problem. The dual of this SDP problem corresponds to adding a simple constraint to the well-known Shor relaxation of
QPS. We show that the max d.c. bound is dominated by another known bound based on a copositive relaxation of QPS, also obtainable
via SDP at comparable computational expense. We also discuss extensions of the d.c. bound to more general quadratic programming
problems. For the application of QPS to bounding the stability number of a graph, we use a novel formulation of the Lovasz
ϑ number to compare ϑ, Schrijver’s ϑ′, and the max d.c. bound. 相似文献
15.
We introduce a new upper bound for the maximum-entropy sampling problem. Our bound is described as the solution of a linear
integer program. The bound depends on a partition of the underlying set of random variables. For the case in which each block
of the partition has bounded cardinality, we describe an efficient dynamic-programming algorithm to calculate the bound. For
the very special case in which the blocks have no more than two elements, we describe an efficient algorithm for calculating
the bound based on maximum-weight matching. This latter formulation has particular value for local-search procedures that
seek to find a good partition. We relate our bound to recent bounds of Hoffman, Lee and Williams. Finally, we report on the
results of some computational experiments.
Received: September 27, 2000 / Accepted: July 26, 2001 Published online: September 5, 2002
Key words. experimental design – design of experiments – entropy – maximum-entropy sampling – matching – integer program – spectral
bound – Fischer's inequality – branch-and-bound – dynamic programming
Mathematics Subject Classification (2000): 52B12, 90C10
Send offprint requests to: Jon Lee
Correspondence to: Jon Lee 相似文献
16.
We define the timetable constrained distance minimization problem (TCDMP) which is a sports scheduling problem applicable for tournaments where the total travel distance must be minimized.
The problem consists of finding an optimal home-away assignment when the opponents of each team in each time slot are given.
We present an integer programming, a constraint programming formulation and describe two alternative solution methods: a hybrid
integer programming/constraint programming approach and a branch and price algorithm. We test all four solution methods on
benchmark problems and compare the performance. Furthermore, we present a new heuristic solution method called the circular traveling salesman approach (CTSA) for solving the traveling tournament problem. The solution method is able to obtain high quality solutions almost instantaneously, and by applying the TCDMP, we show
how the solutions can be further improved. 相似文献
17.
We present a new equivalence result between restricted b‐factors in bipartite graphs and combinatorial t‐designs. This result is useful in the construction of t‐designs by polyhedral methods. We propose a novel linear integer programming formulation, which we call GDP, for the problem of finding t‐designs that has a noteworthy advantage compared to the traditional set‐covering formulation. We analyze some polyhedral properties of GPD, implement a branch‐and‐cut algorithm using it and solve several instances of small designs to compare with another point‐block formulation found in the literature. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 169–182, 2006 相似文献
18.
Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included. 相似文献
19.
Parking Capacity and Pricing in Park'n Ride Trips: A Continuous Equilibrium Network Design Problem 总被引:2,自引:0,他引:2
In this paper we consider the problem of designing parking facilities for park'n ride trips. We present a new continuous equilibrium network design problem to decide the capacity and fare of these parking lots at a tactical level. We assume that the parking facilities have already been located and other topological decisions have already been taken.The modeling approach proposed is mathematical programming with equilibrium constraints. In the outer optimization problem, a central Authority evaluates the performance of the transport network for each network design decision. In the inner problem a multimodal traffic assignment with combined modes, formulated as a variational inequality problem, generates the share demand for modes of transportation, and for parking facilities as a function of the design variables of the parking lots. The objective is to make optimal parking investment and pricing decisions in order to minimize the total travel cost in a subnetwork of the multimodal transportation system.We present a new development in model formulation based on the use of generalized parking link cost as a design variable.The bilevel model is solved by a simulated annealing algorithm applied to the continuous and non-negative design decision variables. Numerical tests are reported in order to illustrate the use of the model, and the ability of the approach to solve applications of moderate size. 相似文献
20.
Over the last decade, there has been increased attention to closed-loop logistics networks. Environmental legislation requires companies to be more responsible by collecting used products from customers. Companies can also benefit from savings that are related to recovering and recycling used products. Unlike previous studies, which only consider single products or a single period of time in multi-objective problems, this paper considers a multi-product multi-period closed-loop logistics network with different types of facilities. A?multi-objective mixed-integer nonlinear programming formulation is developed to minimize the total cost, the delivery time of new products, and the collection time of used products. Thus, this model better approximates real-life applications of closed-loop logistics problems. Interactive fuzzy goal programming (IFGP) is applied to solve the model for handling multiple objective problems with conflicting objectives and to address the imprecise nature of decision-makers?? aspiration levels for goals. The results from computational experiments performed here show that by changing the upper or lower bound of each objective function, one can obtain a better final solution of the problem and also can provide more options for decision makers to choose from based on their situation. Finally, the utilization rate of facilities is shown to be an important indicator when designing a logistics network. 相似文献