An Integer Linear Programming Formulation and Branch-and-Cut Algorithm for the Capacitated m-Ring-Star Problem

We study the capacitated m-ring-star problem (CmRSP) that faces the design of minimum cost network structure that connects customers with m rings using a set of ring connections that share a distinguished node (depot), and optionally star connections that connect customers to ring nodes. Ring and star connections have some associated costs. Also, rings can include transit nodes, named Steiner nodes, to reduce the total network cost if possible. The number of customers in each ring-star (ringʼs customers and customer connected to it through star connections) have an upper bound (capacity).These kind of networks are appropriate in optical fiber urban environments. CmRSP is know to be NP-Hard. In this paper we propose an integer linear programming formulation and a branch-and-cut algorithm.     

Heuristic algorithms for the multi-depot ring-star problem

In this paper, we consider the problem of designing urban optical networks. In particular, given a set of telephone exchanges, we must design a collection of ring-stars, where each ring-star is a cycle composed of a telephone exchange, some customers, some transition points used to save routing costs and customers not on the cycle connected to the cycle by a single edge. The ring topology is chosen in many fiber optic communication networks since it allows to prevent the loss of connection due to a single edge or even a single node failure. The objective is to minimize the total cost of the optical network which is mainly due to the excavation costs. We call this problem Multi-Depot Ring-Star Problem (MDRSP) and we formulate it as an optimization problem in Graph Theory. We present lower bounds and heuristic algorithms for the MDRSP. Computational results on randomly generated instances and real-life datasets are also presented.     

Enhanced formulations and branch-and-cut for the two level network design problem with transition facilities

We consider a new combinatorial optimization problem that combines network design and facility location aspects. Given a graph with two types of customers and two technologies that can be installed on the edges, the objective is to find a minimum cost subtree connecting all customers while the primary customers are served by a primary subtree that is embedded into the secondary subtree. In addition, besides fixed link installation costs, facility opening costs, associated to each node where primary and secondary subtree connect, have to be paid. The problem is called the Two Level Network Design Problem with Transition Facilities (TLNDF).     

The complete optimal stars-clustering-tree problem

We consider the following complete optimal stars-clustering-tree problem: Given a complete graph G=(V,E) with a weight on every edge and a collection of subsets of V, we want to find a minimum weight spanning tree T such that each subset of the vertices in the collection induces a complete star in T. One motivation for this problem is to construct a minimum cost (weight) communication tree network for a collection of (not necessarily disjoint) groups of customers such that each group induces a complete star. As a result the network will provide a “group broadcast” property, “group fault tolerance” and “group privacy”. We present another motivation from database systems with replications. For the case where the intersection graph of the subsets is connected we present a structure theorem that describes all feasible solutions. Based on it we provide a polynomial algorithm for finding an optimal solution. For the case where each subset induces a complete star minus at most k leaves we prove that the problem is NP-hard.     

Single vehicle routing problems with a predefined customer sequence, compartmentalized load and stochastic demands

We consider the problem of finding the optimal routing of a single vehicle that delivers K different products to N customers according to a particular customer order. The demands of the customers for each product are assumed to be random variables with known distributions. Each product type is stored in its dedicated compartment in the vehicle. Using a suitable dynamic programming algorithm we find the policy that satisfies the demands of the customers with the minimum total expected cost. We also prove that this policy has a specific threshold-type structure. Furthermore, we investigate a corresponding infinite-time horizon problem in which the service of the customers does not stop when the last customer has been serviced but it continues indefinitely with the same customer order. It is assumed that the demands of the customers at different tours have the same distributions. It is shown that the discounted-cost optimal policy and the average-cost optimal policy have the same threshold-type structure as the optimal policy in the original problem. The theoretical results are illustrated by numerical examples.     

A linear time algorithm for inverse obnoxious center location problems on networks

For an inverse obnoxious center location problem, the edge lengths of the underlying network have to be changed within given bounds at minimum total cost such that a predetermined point of the network becomes an obnoxious center location under the new edge lengths. The cost is proportional to the increase or decrease, resp., of the edge length. The total cost is defined as sum of all cost incurred by length changes. For solving this problem on a network with m edges an algorithm with running time ${\mathcal{O}(m)}$ is developed.     

Inverse minimum cost flow problems under the weighted Hamming distance

Given a network N(V, A, u, c) and a feasible flow x⁰, an inverse minimum cost flow problem is to modify the cost vector as little as possible to make x⁰ form a minimum cost flow of the network. The modification can be measured by different norms. In this paper, we consider the inverse minimum cost flow problems, where the modification of the arcs is measured by the weighted Hamming distance. Both the sum-type and the bottleneck-type cases are considered. For the former, it is shown to be APX-hard due to the weighted feedback arc set problem. For the latter, we present a strongly polynomial algorithm which can be done in O(n · m²).     

Solving the Passive Optical Network with Fiber Duct Sharing Planning Problem Using Discrete Techniques

Similar to the constrained facility location problem, the passive optical network (PON) planning problem necessitates the search for a subset of deployed facilities (splitters) and their allocated demand points (optical network units) to minimize the overall deployment cost. In this paper we use a mixed integer linear programming formulation stemming from network flow optimization to construct a heuristic based on limiting the total number of interconnecting paths when implementing fiber duct sharing. Then, a disintegration heuristic involving the construction of valid clusters from the output of a k means algorithm, reduce the time complexity while ensuring close to optimal results. The proposed heuristics are then evaluated using a real-world dataset, showing favourable performance.     

Deterministic flow-demand location problems

In flow-covering (interception) models the focus is on the demand for service that originates from customers travelling in the network (not for the purpose of obtaining the service). In contrast, in traditional location models a central assumption is that the demand for service comes from customers residing at nodes of the network. In this paper we combine these two types of models. The paper presents four new problems. Two of the four deal with the problem of locating m facilities so as to maximize the total number of potential customers covered by the facilities (where coverage does not necessarily imply the actual consumption of service). In the two other problems the attention is directed to the consumption of service and thus the criteria is to maximize (minimize) the number of actual users (distance travelled). It is shown in the paper that all four problems have similar structure to other known location problems.     

Efficient heuristics for the rectilinear distance capacitated multi-facility Weber problem

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.     

A location–allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations

The capacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane, and allocating their capacities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where the customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation, which is even more difficult than its deterministic version. We then propose an alternate location–allocation local search heuristic generalizing the ideas used originally for the deterministic problem. In its original form, the applicability of the heuristic depends on the calculation of the expected distances between the facilities and customers, which can be done for only very few distance and probability density function combinations. We therefore propose approximation methods which make the method applicable for any distance function and bivariate location distribution.     

Optimization and sensitivity analysis of controlling arrivals in the queueing system with single working vacation

This paper analyzes the F-policy M/M/1/K queueing system with working vacation and an exponential startup time. The F-policy deals with the issue of controlling arrivals to a queueing system, and the server requires a startup time before allowing customers to enter the system. For the queueing systems with working vacation, the server can still provide service to customers rather than completely stop the service during a vacation period. The matrix-analytic method is applied to develop the steady-state probabilities, and then obtain several system characteristics. We construct the expected cost function and formulate an optimization problem to find the minimum cost. The direct search method and Quasi-Newton method are implemented to determine the optimal system capacity K, the optimal threshold F and the optimal service rates (μ_B,μ_V) at the minimum cost. A sensitivity analysis is conducted to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical examples are provided for illustration purpose.     

Solving the uncapacitated multi-facility Weber problem by vector quantization and self-organizing maps

The uncapacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane and allocating the demands of n customers to these facilities with the minimum total transportation cost. This is a non-convex optimization problem and difficult to solve exactly. As a consequence, efficient and accurate heuristic solution procedures are needed. The problem has different types based on the distance function used to model the distance between the facilities and customers. We concentrate on the rectilinear and Euclidean problems and propose new vector quantization and self-organizing map algorithms. They incorporate the properties of the distance function to their update rules, which makes them different from the existing two neural network methods that use rather ad hoc squared Euclidean metric in their updates even though the problem is originally stated in terms of the rectilinear and Euclidean distances. Computational results on benchmark instances indicate that the new methods are better than the existing ones, both in terms of the solution quality and computation time.     

Dynamic scheduling of a GI/GI/1+GI queue with multiple customer classes

We consider a dynamic control problem for a GI/GI/1+GI queue with multiclass customers. The customer classes are distinguished by their interarrival time, service time, and abandonment time distributions. There is a cost c _k >0 for every class k∈{1,2,…,N} customer that abandons the queue before receiving service. The objective is to minimize average cost by dynamically choosing which customer class the server should next serve each time the server becomes available (and there are waiting customers from at least two classes). It is not possible to solve this control problem exactly, and so we formulate an approximating Brownian control problem. The Brownian control problem incorporates the entire abandonment distribution of each customer class. We solve the Brownian control problem under the assumption that the abandonment distribution for each customer class has an increasing failure rate. We then interpret the solution to the Brownian control problem as a control for the original dynamic scheduling problem. Finally, we perform a simulation study to demonstrate the effectiveness of our proposed control.     

Inverse 1-median problem on trees under weighted Hamming distance

The inverse 1-median problem consists in modifying the weights of the customers at minimum cost such that a prespecified supplier becomes the 1-median of modified location problem. A linear time algorithm is first proposed for the inverse problem under weighted l _?? norm. Then two polynomial time algorithms with time complexities O(n log n) and O(n) are given for the problem under weighted bottleneck-Hamming distance, where n is the number of vertices. Finally, the problem under weighted sum-Hamming distance is shown to be equivalent to a 0-1 knapsack problem, and hence is ${\mathcal{NP}}$ -hard.     

Modeling fuzzy capacitated p-hub center problem and a genetic algorithm solution

Hub and spoke networks are used to switch and transfer commodities between terminal nodes in distribution systems at minimum cost and/or time. The p-hub center allocation problem is to minimize maximum travel time in networks by locating p hubs from a set of candidate hub locations and allocating demand and supply nodes to hubs. The capacities of the hubs are given. In previous studies, authors usually considered only quantitative parameters such as cost and time to find the optimum location. But it seems not to be sufficient and often the critical role of qualitative parameters like quality of service, zone traffic, environmental issues, capability for development in the future and etc. that are critical for decision makers (DMs), have not been incorporated into models. In many real world situations qualitative parameters are as much important as quantitative ones. We present a hybrid approach to the p-hub center problem in which the location of hub facilities is determined by both parameters simultaneously. Dealing with qualitative and uncertain data, Fuzzy systems are used to cope with these conditions and they are used as the basis of this work. We use fuzzy VIKOR to model a hybrid solution to the hub location problem. Results are used by a genetic algorithm solution to successfully solve a number of problem instances. Furthermore, this method can be used to take into account more desired quantitative variables other than cost and time, like future market and potential customers easily.     

Comparison of two randomized policy M/G/1 queues with second optional service, server breakdown and startup

The problem addressed in this paper is to compare the minimum cost of the two randomized control policies in the M/G/1 queueing system with an unreliable server, a second optional service, and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service until the system becomes empty. After all customers are served in the queue, the server immediately takes a vacation and the system operates the (T, p)-policy or (p, N)-policy. For those two policies, the expected cost functions are established to determine the joint optimal threshold values of (T, p) and (p, N), respectively. In addition, we obtain the explicit closed form of the joint optimal solutions for those two policies. Based on the minimal cost, we show that the optimal (p, N)-policy indeed outperforms the optimal (T, p)-policy. Numerical examples are also presented for illustrative purposes.     

Set covering location models with stochastic critical distances

This paper formulates a new version of set covering models by introducing a customer-determined stochastic critical distance. In this model, all services are provided at the sites of facilities, and customers have to go to the facility sites to obtain the services. Due to the randomness of their critical distance, customers patronize a far or near facility with a probability. The objective is to find a minimum cost set of facilities so that every customer is covered by at least one facility with an average probability greater than a given level α. We consider an instance of the problem by embedding the exponential effect of distance into the model. An algorithm based on two searching paths is proposed for solutions to the instance. Experiments show that the algorithm performs well for problems with greater α, and the experimental results for smaller α are reported and analysed.     

Time-cost trade-off analysis of project networks in fuzzy environments

This paper proposes a novel approach for time-cost trade-off analysis of a project network in fuzzy environments. Different from the results of previous studies, in this paper the membership function of the fuzzy minimum total crash cost is constructed based on Zadeh’s extension principle and fuzzy solutions are provided. A pair of two-level mathematical programs parameterized by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy minimum total crash cost at α. By enumerating different values of α, the membership function of the fuzzy minimum total crash cost is constructed, and the corresponding optimal activity time for each activity is also obtained at the same time. An example of time-cost trade-off problem with several fuzzy parameters is solved successfully to demonstrate the validity of the proposed approach. Since the minimum total crash cost is expressed by a membership function rather than by a crisp value, the fuzziness of parameters is conserved completely, and more information is provided for time-cost trade-off analysis in project management. The proposed approach also can be applied to time-cost trade-off problems with other characteristics.     

Improving multicut in directed trees by upgrading nodes

In this paper, we consider the network improvement problem for multicut by upgrading nodes in a directed tree T = (V, E) with multiple sources and multiple terminals. In a node based upgrading model, a node v can be upgraded at the expense of c(v) and such an upgrade reduces weights on all edges incident to v. The objective is to upgrade a minimum cost subset S ⊆ V of nodes such that the resulting network has a multicut in which no edge has weight larger than a given value D. We first obtain a minimum cardinality node multicut V_c for tree T, then find the minimum cost upgrading set based on the upgrading sets for the subtrees rooted at the nodes in V_c. We show that our algorithm is polynomial when the number of source–terminal pairs is upper bounded by a given value.