An optimal capacity assignment for the robust design problem in capacitated flow networks

This paper proposes an exact algorithm to solve the robust design problem in a capacitated flow network in which each edge has several possible capacities. A capacitated flow network is popular in our daily life. For example, the computer network, the power transmission network, or even the supply chain network are capacitated flow networks. In practice, such network may suffer failure, partial failure or maintenance. Therefore, each edge in the network should be assigned sufficient capacity to keep the network functioning normally. The robust design problem (RDP) in a capacitated flow network is to search for the minimum capacity assignment of each edge such that the network still survived even under the edge’s failure. However, how to optimally assign the capacity to each edge is not an easy task. Although this kind of problem was known of NP-hard, this paper proposes an efficient exact algorithm to search for the optimal solutions for such a network and illustrates the efficiency of the proposed algorithm by numerical examples.     

Multistate components assignment problem with optimal network reliability subject to assignment budget

The typical assignment problem for finding the optimal assignment of a set of components to a set of locations in a system has been widely studied in practical applications. However, this problem mainly focuses on maximizing the total profit or minimizing the total cost without considering component’s failure. In practice, each component should be multistate due to failure, partially failure, or maintenance. That is, each component has several capacities with a probability distribution and may fail. When a set of multistate components is assigned to a system, the system can be treated as a stochastic-flow network. The network reliability is the probability that d units of homogenous commodity can be transmitted through the network successfully. The multistate components assignment problem to maximize the network reliability is never discussed. Therefore, this paper focuses on solving this problem under an assignment budget constraint, in which each component has an assignment cost. The network reliability under a components assignment can be evaluated in terms of minimal paths and state-space decomposition. Subsequently an optimization method based on genetic algorithm is proposed. The experimental results show that the proposed algorithm can be executed in a reasonable time.     

Single-commodity robust network design problem: Complexity,instances and heuristic solutions

We study a single-commodity Robust Network Design problem (RND) in which an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. In each scenario, a subset of the nodes is exchanging flow. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. Previously conducted computational investigations on the problem motivated the study of the complexity of some special cases and we present complexity results on them, including hypercubes. In turn, these results lead to the definition of new instances (random graphs with {−1, 0, 1} balances) that are computationally hard for the natural flow formulation. These instances are then solved by means of a new heuristic algorithm for RND, which consists of three phases. In the first phase the graph representing the network is reduced by heuristically deleting a subset of the arcs, and a feasible solution is built. The second phase consists of a neighborhood search on the reduced graph based on a Mixed-Integer (Linear) Programming (MIP) flow model. Finally, the third phase applies a proximity search approach to further improve the solution, taking into account the original graph. The heuristic is tested on the new instances, and the comparison with the solutions obtained by Cplex on a natural flow formulation shows the effectiveness of the proposed method.     

A cardinality-constrained robust model for the assignment problem in Home Care services

Home Care includes medical, paramedical and social services which are delivered to patients at their domicile rather than in hospital. Managing human and material resources in Home Care services is a difficult task, as the provider has to deal with peculiar constraints (e.g., the continuity of care, which imposes that a patient is always cared for by the same nurse) and to manage the high variability of patients’ demands. One of the main issues encountered in planning Home Care services under continuity of care requirement is the nurse-to-patient assignment. Despite the importance of this topic, the problem is only marginally addressed in the literature, where continuity of care is usually treated as a soft-constraint rather than as a hard one. Uncertainty is another relevant feature of nurse-to-patient assignment problem, and it is usually managed adopting stochastic programming or analytical policies. However, both these approaches proved to be limited, even if they improve the quality of the assignments upon those actually provided in practice. In this paper, we develop a cardinality-constrained robust assignment model, which allows exploiting the potentialities of a mathematical programming model without the necessity of generating scenarios. The developed model is tested on real-life instances related to a relevant Home Care provider operating in Italy, in order to evaluate its capability of reducing the costs related to nurses’ overtimes.     

The design of robust value-creating supply chain networks: A critical review

This paper discusses Supply Chain Network (SCN) design problem under uncertainty, and presents a critical review of the optimization models proposed in the literature. Some drawbacks and missing aspects in the literature are pointed out, thus motivating the development of a comprehensive SCN design methodology. Through an analysis of supply chains uncertainty sources and risk exposures, the paper reviews key random environmental factors and discusses the nature of major disruptive events threatening SCN. It also discusses relevant strategic SCN design evaluation criteria, and it reviews their use in existing models. We argue for the assessment of SCN robustness as a necessary condition to ensure sustainable value creation. Several definitions of robustness, responsiveness and resilience are reviewed, and the importance of these concepts for SCN design is discussed. This paper contributes to framing the foundations for a robust SCN design methodology.     

Probabilistic analysis of the generalised assignment problem

We analyse thegeneralised assignment problem under the assumption that all coefficients are drawn uniformly and independently from [0, 1]. We show that such problems can be solved exactly with high probability, in a well-defined sense. The results are closely related to earlier work of Lueker, Goldberg and Marchetti-Spaccamela and ourselves.Supported by NATO grant RG0088/89.Supported by NSF grant CCR-8900112 and NATO grant RG0088/89.     

A simple dual algorithm for the generalised assignment problem

A new algorithm for the generalised assignment problem is described in this paper. The dual-type algorithm uses a simple heuristic derived from a relaxation of the problem. The algorithm has been tested on generalised assignment problems of substantial size and compared to an exact integer programming approach and a well-established heuristic approach. Computational results look promising in terms of speed and solution quality.     

An analysis of the structural complexity of supply chain networks

Each enterprise in a supply chain network needs quantitative indicators to analyze and manage its interactions with different business partners in the network. Supply chains exhibit the characteristics of complex systems. In a supply chain network, a large number of firms cooperate simultaneously with many suppliers and customers, and interact through a variety of information and material flows to achieve a balance between supply and demand. However, the complexity of a supply chain is not a simple linear structure where a small change often results in a chain reaction. When supply chain complexity increases, monitoring and managing the interaction between different elements of the chain becomes more difficult. An entropy model based on information theory provides an appropriate means of quantifying the complexity of a supply chain system by delivering information required to describe the state of the system. The entropy measure links uncertainty and complexity so that, as a system grows in uncertainty, it becomes more complex and more information is required to describe and monitor it. In this paper, we propose an entropy-based measure for analyzing the structural complexity in relation to the structure and system uncertainty. The method provides guidelines for estimating the complexity throughout the supply chain structure.     

Models and algorithms for the optimization of signal settings on urban networks with stochastic assignment models

In this paper models and algorithms for the optimization of signal settings on urban networks are proposed. Two different approaches to the solution of the problem may be identified: a global approach (optimization of intersection signal settings on the whole network) and a local approach (optimization of signal settings intersection by intersection). For each approach a different optimization model and some solution algorithms are proposed; both models and algorithms are based on the assumptions of within-day static system and stochastic user equilibrium assignment models. The paper includes numerical results on test networks and a comparison between the two approaches.     

Compounded genetic algorithms for the quadratic assignment problem

We introduce the compounded genetic algorithm. We propose to run a quick genetic algorithm several times as Phase 1, and compile the best solutions in each run to create a starting population for Phase 2. This new approach was tested on the quadratic assignment problem with very good results.     

Lower bounds for the quadratic assignment problem

We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branch-and-bound type algorithm for the quadratic assignment problem.     

A survey of algorithms for the generalized assignment problem

This paper surveys algorithms for the well-known problem of finding the minimum cost assignment of jobs to agents so that each job is assigned exactly once and agents are not overloaded. All approaches seem to be based on branch-and-bound with bound supplied through heuristics and through relaxations of the primal problem formulation. From the survey one can select building blocks for the design of one's own tailor-made algorithm. The survey also reveals that although just about every mathematical programming technique was tried on this problem, there is still a lack of a representative set of test problems on which competing enumeration algorithms can be compared, as well as a shortage of effective heuristics.     

Lower bounds for the axial three-index assignment problem

This paper describes new bounding methods for the axial three-index assignment problem (3AP). For calculating 3AP lower bounds, we use a projection method followed by a Hungarian algorithm, based on a new Lagrangian relaxation. We also use a cost transformation scheme, which iteratively transforms 3AP costs in a series of equivalent 3APs, which provides the possibility of improving the 3AP lower bound. These methods produce efficiently computed relatively tight lower bound.     

An average case analysis of the minimum spanning tree heuristic for the power assignment problem

We present an average case analysis of the minimum spanning tree heuristic for the power assignment problem. The worst‐case approximation ratio of this heuristic is 2. We show that in Euclidean d‐dimensional space, when the vertex set consists of a set of i.i.d. uniform random independent, identically distributed random variables in [0,1]^d, and the distance power gradient equals the dimension d, the minimum spanning tree‐based power assignment converges completely to a constant depending only on d.     

A simplex-based labelling algorithm for the linear fractional assignment problem

This paper constructs an algorithm to solve the fractional assignment problem. Algorithms that are currently used are mostly based on parametric approaches and must solve a sequence of optimization procedures. They also neglect the difficulties caused by degeneracy. The proposed algorithm performs optimization once and overcomes degeneracy. The main features of the algorithm are an effective initial heuristic approach, a simple labelling procedure and an implicit primal-dual schema. A numerical example is presented and demonstrates that the proposed algorithm is easy to apply. Computational results are compared with those from other developed methods. The results show that the proposed algorithm is efficient.     

Lower bounds for the quadratic assignment problem via triangle decompositions

We consider transformations of the (metric) Quadratic Assignment Problem (QAP) that exploit the metric structure of a given instance. We show in particular how the structural properties of rectangular grids can be used to improve a given lower bound. Our work is motivated by previous research of Palubetskes (1988), and it extends a bounding approach proposed by Chakrapani and Skorin-Kapov (1993). Our computational results indicate that the present approach is practical; it has been applied to problems of dimension up ton = 150. Moreover, the new approach yields by far the best lower bounds on most of the instances of metric QAPs that we considered.The authors gratefully acknowledge financial support by the Christian Doppler Laboratorium für Diskrete Optimierung.     

Tabu Search for the planar three-index assignment problem

Tabu Search is a very effective method for approximately solving hard combinatorial problems. In the current work we implement Tabu Search for solving the Planar Three-Index Assignment Problem. The problem deals with finding the minimum cost assignment between elements of three distinct sets demanding that every pair of elements, each representing a different set, appears in the same solution exactly once. The solutions of the problems correspond to Latin squares. These structures form the basis of the move generation mechanism employed by the Tabu Search procedures. Standard Tabu Search ideas such as candidate move list, variable tabu list size, and frequency-based memory are tested. Computational results for a range of problems of varying dimensions are presented.     

An approximation algorithm for the generalized assignment problem

The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei requires timep _ij and incurs a cost ofc _ij ; each machinei is available forT _i time units, and the objective is to minimize the total cost incurred. Our main result is as follows. There is a polynomial-time algorithm that, given a valueC, either proves that no feasible schedule of costC exists, or else finds a schedule of cost at mostC where each machinei is used for at most 2T _i time units.We also extend this result to a variant of the problem where, instead of a fixed processing timep _ij , there is a range of possible processing times for each machine—job pair, and the cost linearly increases as the processing time decreases. We show that these results imply a polynomial-time 2-approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomial-time algorithm that, given valuesM andT, either proves that no schedule of mean job completion timeM and makespanT exists, or else finds a schedule of mean job completion time at mostM and makespan at most 2T. Research partially supported by an NSF PYI award CCR-89-96272 with matching support from UPS, and Sun Microsystems, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.Research supported in part by a Packard Fellowship, a Sloan Fellowship, an NSF PYI award, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

Autocorrelation measures for the quadratic assignment problem

In this article we provide an exact expression for computing the autocorrelation coefficient ξ and the autocorrelation length ? of any arbitrary instance of the Quadratic Assignment Problem (QAP) in polynomial time using its elementary landscape decomposition. We also provide empirical evidence of the autocorrelation length conjecture in QAP and compute the parameters ξ and ? for the 137 instances of the QAPLIB. Our goal is to better characterize the difficulty of this important class of problems to ease the future definition of new optimization methods. Also, the advance that this represents helps to consolidate QAP as an interesting and now better understood problem.