An effective algorithm for the two-stage location problem on a tree-like network

A two-stage facility location problem on a tree-like network is considered under the restriction that the transportation costs for a unit of production from one node to another is equal to the sum of the edges in the path connecting these nodes. Some exact algorithm with time complexity O(nm ³) is suggested for this problem, where n is the number of the production demand points and, m is an upper bound on the number of possible facility location sites of each stage.     

Concave cost minimization on networks

The paper deals with the problem of finding a minimum cost multicommodity flow on an uncapacitated network with concave link costs. Problems of this type are the optimal design of a network in the presence of scale economies and the telpack problem.Two different definitions of local optimality are given and compared both from the point of view of the computational complexity and from the point of view of the goodness of the solution they may provide.A vertex following algorithm to find a local optimum is proposed. The computational complexity of each iteration of the algorithm is O(n³), where n is the number of nodes of the network, and is independent of the differentiability of the objective function.Experimental results obtained from a set of test problems of size ranging from 11 nodes and 23 arcs to 48 nodes and 174 arcs, with number of commodities up to 5, are given.     

Optimal M/G/1 Server Location on a Network Having a Fixed Facility

This paper considers the problem of locating a single mobile service unit on a network G where the servicing of a demand includes travel time to a permanent facility which is located at a predetermined point on G. Demands for service, which occur solely on the nodes of the network, arrive in a homogeneous Poisson manner. The server, when free, can be immediately dispatched to a demand: the service unit travels to the demand, performs some on-scene service, continues to the permanent facility, where off-scene service is rendered, and then it returns to its ‘home’ location, where possibly additional off-scene service is given. Previous research has examined the same problem, however without the presence of a permanent facility. The paper discusses methods of solving two cases when the server is unable to be immediately dispatched to service a demand: (1) the zero-capacity queueing system; (2) the infinite-capacity queueing system. For the first case we prove that the optimal location is included in a small set of points in the network, and we show how to find this set. For the second case, we present an 0(n³) algorithm (n is the number of nodes) to obtain the optimal location.     

The Optimum Location of Multi-centres on a Graph

The location of a number of service centres on a network (graph) in such a way so that every node (demand point) lies within a critical distance of at least one of the centres appears often in problems involving emergency services. When the number p of centres is fixed and what is required is their location so that this critical distance is the smallest possible, the resulting location is called "the absolute p-centre of the graph". This paper presents an iterative algorithm for finding absolute p-centres in general (weighted or unweighted) graphs. The algorithm is shown to be computationally efficient for quite large graphs. Results (computing times and numbers of iterations) are given for 15 test graphs varying in size from 10 to 50 nodes and from 20 to 120 links.     

Relaxed voting and competitive location under monotonous gain functions on trees

We examine competitive location problems where two competitors serve a good to users located in a network. Users decide for one of the competitors based on the distance induced by an underlying tree graph. The competitors place their server sequentially into the network. The goal of each competitor is to maximize his benefit which depends on the total user demand served. Typical competitive location problems include the (1,X₁)-medianoid, the (1,1)-centroid, and the Stackelberg location problem.An additional relaxation parameter introduces a robustness of the model against small changes in distance. We introduce monotonous gain functions as a general framework to describe the above competitive location problems as well as several problems from the area of voting location such as Simpson, Condorcet, security, and plurality.In this paper we provide a linear running time algorithm for determining an absolute solution in a tree where competitors are allowed to place on nodes or on inner points. Furthermore we discuss the application of our approach to the discrete case.     

Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

This paper presents a methodology to find near-optimal joint inventory control policies for the real case of a one-warehouse, n-retailer distribution system of infusion solutions at a University Medical Center in France. We consider stochastic demand, batching and order-up-to level policies as well as aspects particular to the healthcare setting such as emergency deliveries, required service level rates and a new constraint on the ordering policy that fits best the hospital’s interests instead of abstract ordering costs. The system is modeled as a Markov chain with an objective to minimize the stock-on-hand value for the overall system. We provide the analytical structure of the model to show that the optimal reorder point of the policy at both echelons is easily derived from a simple probability calculation. We also show that the optimal policy at the care units is to set the order-up-to level one unit higher than the reorder point. We further demonstrate that optimizing the care units in isolation is optimal for the joint multi-echelon, n-retailer problem. A heuristic algorithm is presented to find the near-optimal order-up-to level of the policy of each product at the central pharmacy; all other policy parameters are guaranteed optimal via the structure provided by the model. Comparison of our methodology versus that currently in place at the hospital showed a reduction of approximately 45% in the stock-on-hand value while still respecting the service level requirements.     

A polynomial time primal network simplex algorithm for minimum cost flows

Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n ²m lognC, n ²m² logn)) time, wheren is the number of nodes in the network,m is the number of arcs, andC denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the “premultiplier algorithm”. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm lognC, nm ² logn)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm logn).     

Approximate KKT points and a proximity measure for termination

Karush–Kuhn–Tucker (KKT) optimality conditions are often checked for investigating whether a solution obtained by an optimization algorithm is a likely candidate for the optimum. In this study, we report that although the KKT conditions must all be satisfied at the optimal point, the extent of violation of KKT conditions at points arbitrarily close to the KKT point is not smooth, thereby making the KKT conditions difficult to use directly to evaluate the performance of an optimization algorithm. This happens due to the requirement of complimentary slackness condition associated with KKT optimality conditions. To overcome this difficulty, we define modified ${\epsilon}$ -KKT points by relaxing the complimentary slackness and equilibrium equations of KKT conditions and suggest a KKT-proximity measure, that is shown to reduce sequentially to zero as the iterates approach the KKT point. Besides the theoretical development defining the modified ${\epsilon}$ -KKT point, we present extensive computer simulations of the proposed methodology on a set of iterates obtained through an evolutionary optimization algorithm to illustrate the working of our proposed procedure on smooth and non-smooth problems. The results indicate that the proposed KKT-proximity measure can be used as a termination condition to optimization algorithms. As a by-product, the method helps to find Lagrange multipliers correspond to near-optimal solutions which can be of importance to practitioners. We also provide a comparison of our KKT-proximity measure with the stopping criterion used in popular commercial softwares.     

Gradient-constrained discounted Steiner trees II: optimally locating a discounted Steiner point

A gradient-constrained discounted Steiner tree is a network interconnecting given set of nodes in Euclidean space where the gradients of the edges are all no more than an upper bound which defines the maximum gradient. In such a tree, the costs are associated with its edges and values are associated with nodes and are discounted over time. In this paper, we study the problem of optimally locating a single Steiner point in the presence of the gradient constraint in a tree so as to maximize the sum of all the discounted cash flows, known as the net present value (NPV). An edge in the tree is labelled as a b edge, or a m edge, or an f edge if the gradient between its endpoints is greater than, or equal to, or less than the maximum gradient respectively. The set of edge labels at a discounted Steiner point is called its labelling. The optimal location of the discounted Steiner point is obtained for the labellings that can occur in a gradient-constrained discounted Steiner tree. In this paper, we propose the gradient-constrained discounted Steiner point algorithm to optimally locate the discounted Steiner point in the presence of a gradient constraint in a network. This algorithm is applied to a case study. This problem occurs in underground mining, where we focus on the optimization of underground mine access to obtain maximum NPV in the presence of a gradient constraint. The gradient constraint defines the navigability conditions for trucks along the underground tunnels.     

An efficient label setting/correcting shortest path algorithm

We design a new label shortest path algorithm by applying the concept of a pseudo permanent label. This approach allows an algorithm to partition the set of nodes into two new sets: pseudo permanently labeled nodes and its complementary set. From this point of view, this new label method can be considered as a label setting method. Moreover, at least one node becomes permanently labeled when some nodes which belong to the set of pseudo permanently labeled nodes are scanned in each iteration of the algorithm. In the case of networks with non-negative length arcs it is easy to prove that this node has the minimum distance label among the non-pseudo permanently labeled nodes. On the other hand, it is not known during the computation which pseudo permanently labeled nodes are permanently labeled. Therefore, all distance labels are temporary and the algorithm becomes a label correcting method. Nevertheless, the proposed algorithm exhibits some nice features, such as: (1) the time bound for the running of the algorithm for a network with n nodes and m arcs is O(nm); (2) the number of node scan operations in the algorithm is less than the number of these operations in the previous label correcting algorithms as is observed in the computational experience; (3) the algorithm incorporates two new rules which allow easy detection of a negative cycle in the network; (4) the algorithm is quite simple and very easy to implement, and does not require sophisticated data structures; (5) the algorithm exhibits flexibility in the order in which the new pseudo permanently labeled nodes are scanned. The above features are possible through the application of the pseudo permanent label concept.     

Efficient contraflow algorithms for quickest evacuation planning

The optimization models and algorithms with their implementations on flow over time problems have been an emerging field of research because of largely increasing human-created and natural disasters worldwide. For an optimal use of transportation network to shift affected people and normalize the disastrous situation as quickly and Efficiently as possible, contraflow configuration is one of the highly applicable operations research (OR) models. It increases the outbound road capacities by reversing the direction of arcs towards the safe destinations that not only minimize the congestion and increase the flow but also decrease the evacuation time significantly. In this paper, we sketch the state of quickest flow solutions and solve the quickest contraflow problem with constant transit times on arcs proving that the problem can be solved in strongly polynomial time O(nm²(log n)²), where n and m are number of nodes and number of arcs, respectively in the network. This contraflow solution has the same computational time bound as that of the best min-cost flow solution. Moreover, we also introduce the contraflow approach with load dependent transit times on arcs and present an Efficient algorithm to solve the quickest contraflow problem approximately. Supporting the claim, our computational experiments on Kathmandu road network and on randomly generated instances perform very well matching the theoretical results. For sufficiently large number of evacuees, about double flow can be shifted with the same evacuation time and about half time is sufficient to push the given flow value with contraflow reconfiguration.     

Development of the tree-based link labeling algorithm for optimal path-finding in urban transportation networks

The optimal path-finding algorithm which is an important module in developing route guidance systems and traffic control systems has to provide correct paths to consider U-turns, P-turns, and no-left-turns in urban transportation networks.Traditional methods which have been used to consider those regulations on urban transportation networks can be categorized into network representation and algorithmic methods like the vine-building algorithm. First, network representation methods use traditional optimal path-finding algorithms with modifications to the network structure: for example, just adding dummy nodes and links to the existing network allows constraint-search in the network. This method which creates large networks is hard to implement and introduces considerable difficulties in network coding. With the increased number of nodes and links, the memory requirement tremendously increases, which causes the processing speed to slow down. For these reasons, the method has not been widely accepted for incorporating turning regulations in optimal path-finding problems in transportation networks. Second, algorithmic methods, as they are mainly based on the vine-building algorithm, have been suggested for determining optimal path for networks with turn penalties and prohibitions. However, the algorithms, although they nicely reflect the characteristics of urban transportation networks, frequently provide infeasible or suboptimal solutions.The algorithm to be suggested in this research is a method which is basically based on Dijkstra's algorithm [1] and the tree-building algorithm used to construct optimal paths. Unlike the traditional node labeling algorithms which label each node with minimum estimated cost, this algorithm labels each link with minimum estimated cost.Comparison with the vine-building algorithm shows that the solution of the link-labeling algorithm is better than that of the vine-building algorithm which very frequently provides suboptimal solutions. As a result, the algorithm allows turning regulations, while providing an optimal solution within a reasonable time limit.     

The <Emphasis Type="Italic">p</Emphasis>-Centre Problem—Heuristic and Optimal Algorithms

The p-centre problem, or minimax location-allocation problem in location theory terminology, is the following: given n demand points on the plane and a weight associated with each demand point, find p new facilities on the plane that minimize the maximum weighted Euclidean distance between each demand point and its closest new facility. We present two heuristics and an optimal algorithm that solves the problem for a given p in time polynomial in n. Computational results are presented.     

A dynamic programming operator for tour location problems applied to the covering tour problem

This paper presents an evaluation operator for single-trip vehicle routing problems where it is not necessary to visit all the nodes. Such problems are known as Tour Location Problems. The operator, called Selector, is a dynamic programming algorithm that converts a given sequence of nodes into a feasible tour. The operator returns a subsequence of this giant tour which is optimal in terms of length. The procedure is implemented in an adaptive large neighborhood search to solve a specific tour location problem: the Covering Tour Problem. This problem consists in finding a lowest-cost Hamiltonian cycle over a subset of nodes such that nodes outside the tour are within a given distance from a visited node. The metaheuristic proposed is competitive as shown by the quality of results evaluated using the output of a state-of-the-art exact algorithm.     

Minmax regret location–allocation problem on a network under uncertainty

We consider a robust location–allocation problem with uncertainty in demand coefficients. Specifically, for each demand point, only an interval estimate of its demand is known and we consider the problem of determining where to locate a new service when a given fraction of these demand points must be served by the utility. The optimal solution of this problem is determined by the “minimax regret” location, i.e., the point that minimizes the worst-case loss in the objective function that may occur because a decision is made without knowing which state of nature will take place. For the case where the demand points are vertices of a network we show that the robust location–allocation problem can be solved in O(min{p, n − p}n³m) time, where n is the number of demand points, p (p < n) is the fixed number of demand points that must be served by the new service and m is the number of edges of the network.     

GASUB: finding global optima to discrete location problems by a genetic-like algorithm

In many discrete location problems, a given number s of facility locations must be selected from a set of m potential locations, so as to optimize a predetermined fitness function. Most of such problems can be formulated as integer linear optimization problems, but the standard optimizers only are able to find one global optimum. We propose a new genetic-like algorithm, GASUB, which is able to find a predetermined number of global optima, if they exist, for a variety of discrete location problems. In this paper, a performance evaluation of GASUB in terms of its effectiveness (for finding optimal solutions) and efficiency (computational cost) is carried out. GASUB is also compared to MSH, a multi-start substitution method widely used for location problems. Computational experiments with three types of discrete location problems show that GASUB obtains better solutions than MSH. Furthermore, the proposed algorithm finds global optima in all tested problems, which is shown by solving those problems by Xpress-MP, an integer linear programing optimizer (21). Results from testing GASUB with a set of known test problems are also provided.     

Cone contraction and reference point methods for multi-criteria mixed integer optimization

Interactive approaches employing cone contraction for multi-criteria mixed integer optimization are introduced. In each iteration, the decision maker (DM) is asked to give a reference point (new aspiration levels). The subsequent Pareto optimal point is the reference point projected on the set of admissible objective vectors using a suitable scalarizing function. Thereby, the procedures solve a sequence of optimization problems with integer variables. In such a process, the DM provides additional preference information via pair-wise comparisons of Pareto optimal points identified. Using such preference information and assuming a quasiconcave and non-decreasing value function of the DM we restrict the set of admissible objective vectors by excluding subsets, which cannot improve over the solutions already found. The procedures terminate if all Pareto optimal solutions have been either generated or excluded. In this case, the best Pareto point found is an optimal solution. Such convergence is expected in the special case of pure integer optimization; indeed, numerical simulation tests with multi-criteria facility location models and knapsack problems indicate reasonably fast convergence, in particular, under a linear value function. We also propose a procedure to test whether or not a solution is a supported Pareto point (optimal under some linear value function).     

On the expected number of optimal and near-optimal solutions to the Euclidean travelling salesman problem

An algorithm for empirically calculating the expected number of optimal and near-optimal solutions in a random Euclidean travelling salesman problem is presented. The algorithm is based on well known geometric properties of the optimal tour. For problems involving up to 15 points uniformily distributed in the unit square, experiments show this expected number to be extremely small.     

A technique for speeding up the solution of the Lagrangean dual

We propose techniques for the solution of the LP relaxation and the Lagrangean dual in combinatorial optimization and nonlinear programming problems. Our techniques find the optimal solution value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm or the recent algorithm of Vaidya. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Our techniques lead to significant improvements in the theoretical running time compared with previously known methods (interior point methods, Ellipsoid algorithm, Vaidya's algorithm). We use our method to the solution of the LP relaxation and the Langrangean dual of several classical combinatorial problems, like the traveling salesman problem, the vehicle routing problem, the Steiner tree problem, thek-connected problem, multicommodity flows, network design problems, network flow problems with side constraints, facility location problems,K-polymatroid intersection, multiple item capacitated lot sizing problem, and stochastic programming. In all these problems our techniques significantly improve the theoretical running time and yield the fastest way to solve them.     

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.