A norm-relaxed method of feasible directions for finely discretized problems from semi-infinite programming

In this paper, a class of finely discretized Semi-Infinite Programming (SIP) problems is discussed. Combining the idea of the norm-relaxed Method of Feasible Directions (MFD) and the technique of updating discretization index set, we present a new algorithm for solving the Discretized Semi-Infinite (DSI) problems from SIP. At each iteration, the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem, i.e., a quadratic program, and some appropriate constraints are chosen to reduce the computational cost. A high-order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints. Under weak conditions such as Mangasarian–Fromovitz Constraint Qualification (MFCQ), the proposed algorithm possesses weak global convergence. Moreover, the superlinear convergence is obtained under Linearly Independent Constraint Qualification (LICQ) and other assumptions. In the end, some elementary numerical experiments are reported.     

Lower and upper bounds for location-arc routing problems with vehicle capacity constraints

This paper addresses multi-depot location arc routing problems with vehicle capacity constraints. Two mixed integer programming models are presented for single and multi-depot problems. Relaxing these formulations leads to other integer programming models whose solutions provide good lower bounds for the total cost. A powerful insertion heuristic has been developed for solving the underlying capacitated arc routing problem. This heuristic is used together with a novel location–allocation heuristic to solve the problem within a simulated annealing framework. Extensive computational results demonstrate that the proposed algorithm can find high quality solutions. We also show that the potential cost saving resulting from adding location decisions to the capacitated arc routing problem is significant.     

An exact algorithm for the precedence-constrained single-machine scheduling problem

This study proposes an efficient exact algorithm for the precedence-constrained single-machine scheduling problem to minimize total job completion cost where machine idle time is forbidden. The proposed algorithm is based on the SSDP (Successive Sublimation Dynamic Programming) method and is an extension of the authors’ previous algorithms for the problem without precedence constraints. In this method, a lower bound is computed by solving a Lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes. Numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence-constrained total weighted tardiness and total weighted earliness–tardiness problems, and most instances with 100 jobs of the former problem.     

A column generation algorithm for the estimation of origin–destination matrices in congested traffic networks

The general problem of estimating origin–destination (O–D) matrices in congested traffic networks is formulated as a mathematical programme with equilibrium constraints, referred to as the demand adjustment problem (DAP). This approach integrates the O–D matrix estimation and the network equilibrium assignment into one process. In this paper, a column generation algorithm for the DAP is presented. This algorithm iteratively solves a deterministic user equilibrium model for a given O–D matrix and a DAP restricted to the previously generated paths, whose solution generates a new O–D trip matrix estimation. The restricted DAP is formulated via a single level optimization problem. The convergence on local minimum of the proposed algorithm requires only the continuity of the link travel cost functions and the gauges used in the definition of the DAP.     

New convergent heuristics for 0–1 mixed integer programming

Several hybrid methods have recently been proposed for solving 0–1 mixed integer programming problems. Some of these methods are based on the complete exploration of small neighborhoods. In this paper, we present several convergent algorithms that solve a series of small sub-problems generated by exploiting information obtained from a series of relaxations. These algorithms generate a sequence of upper bounds and a sequence of lower bounds around the optimal value. First, the principle of a linear programming-based algorithm is summarized, and several enhancements of this algorithm are presented. Next, new hybrid heuristics that use linear programming and/or mixed integer programming relaxations are proposed. The mixed integer programming (MIP) relaxation diversifies the search process and introduces new constraints in the problem. This MIP relaxation also helps to reduce the gap between the final upper bound and lower bound. Our algorithms improved 14 best-known solutions from a set of 108 available and correlated instances of the 0–1 multidimensional Knapsack problem. Other encouraging results obtained for 0–1 MIP problems are also presented.     

Note on “Inverse minimum cost flow problems under the weighted Hamming distance”

Jiang et al. proposed an algorithm to solve the inverse minimum cost flow problems under the bottleneck-type weighted Hamming distance [Y. Jiang, L. Liu, B. Wuc, E. Yao, Inverse minimum cost flow problems under the weighted Hamming distance, European Journal of Operational Research 207 (2010) 50–54]. In this note, it is shown that their proposed algorithm does not solve correctly the inverse problem in the general case due to some incorrect results in that article. Then, a new algorithm is proposed to solve the inverse problem in strongly polynomial time. The algorithm uses the linear search technique and solves a shortest path problem in each iteration.     

A note on article “A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function”

Kim and Whang use a tolerance approach for solving fuzzy goal programming problems with unbalanced membership functions [J.S. Kim, K. Whang, A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function, European Journal of Operational Research 107 (1998) 614–624]. In this note it is shown that some results in that article are incorrect. The necessary corrections are proposed.     

Incremental analysis for MCDM with an application to group TOPSIS

The study aims to exploit incremental analysis or marginal analysis to overcome the drawbacks of ratio scales utilized in various multi-criteria or multi-attribute decision making (MCDM/MADM) techniques. In the proposed 11-step procedure, multiple criteria of alternatives are first reorganized as two categories – benefits and costs – and decision information will be manipulated separately. The performances of alternatives are then evaluated on their incremental benefit–cost ratio, and the rank can be obtained by applying the group TOPSIS (technique for order preference by similarity to ideal solution) model (Shih et al., 2007). Two representations of cost, i.e., a cost index and utility index, are proposed in the model to better-fit real-world situations. In addition, some considerations on costs and input–output relations are also discussed in order to understand the essentials of incremental analysis. In the final part, a case of robot selection demonstrates the suggested model to be both robust and efficient in a group decision-making environment.     

Optimal compromise solution of multi-objective minimal cost flow problems in fuzzy environment

Ghatee and Hashemi [M. Ghatee, S.M. Hashemi, Ranking function-based solutions of fully fuzzified minimal cost flow problem, Inform. Sci. 177 (2007) 4271–4294] transformed the fuzzy linear programming formulation of fully fuzzy minimal cost flow (FFMCF) problems into crisp linear programming formulation and used it to find the fuzzy optimal solution of balanced FFMCF problems. In this paper, it is pointed out that the method for transforming the fuzzy linear programming formulation into crisp linear programming formulation, used by Ghatee and Hashemi, is not appropriate and a new method is proposed to find the fuzzy optimal solution of multi-objective FFMCF problems. The proposed method can also be used to find the fuzzy optimal solution of single-objective FFMCF problems. To show the application of proposed method in real life problems an existing real life FFMCF problem is solved.     

Generalized minimal cost flow problem in fuzzy nature: An application in bus network planning problem

The aim of minimal cost flow problem (MCFP) in fuzzy nature, which is denoted with FMCFP, is to find the least cost of the shipment of a commodity through a capacitated network in order to satisfy imprecise concepts in supply or demand of network nodes and capacity or cost of network links. Fuzzy supply–demand may arise in real problems, where incomplete statistical data or simulation results are used. Also, variation in the cost or capacity of links is commonly happening. In the present paper, after defining a total order on LR type fuzzy numbers, three models are studied; MCFP with fuzzy costs, MCFP with fuzzy supply–demand and a combination of two cases. For the first model, scaling negative cycle cancelling algorithm, which is a polynomial time algorithm, is proposed. For the second model, “nominal flow” is introduced which provides an efficient scheme for finding fuzzy flow. For the third model, we present an exact and some heuristic methods. Numerical examples are illustrated to demonstrate the efficiency of the proposed schemes. Finally, an application of this viewpoint in bus network planning problem is provided.     

Box-constrained multi-objective optimization: A gradient-like method without “a priori” scalarization

The aim of this paper is the development of an algorithm to find the critical points of a box-constrained multi-objective optimization problem. The proposed algorithm is an interior point method based on suitable directions that play the role of gradient-like directions for the vector objective function. The method does not rely on an “a priori” scalarization and is based on a dynamic system defined by a vector field of descent directions in the considered box. The key tool to define the mentioned vector field is the notion of vector pseudogradient. We prove that the limit points of the solutions of the system satisfy the Karush–Kuhn–Tucker (KKT) first order necessary condition for the box-constrained multi-objective optimization problem. These results allow us to develop an algorithm to solve box-constrained multi-objective optimization problems. Finally, we consider some test problems where we apply the proposed computational method. The numerical experience shows that the algorithm generates an approximation of the local optimal Pareto front representative of all parts of optimal front.     

A fuzzy shortest path with the highest reliability

This paper concentrates on a shortest path problem on a network where arc lengths (costs) are not deterministic numbers, but imprecise ones. Here, costs of the shortest path problem are fuzzy intervals with increasing membership functions, whereas the membership function of the total cost of the shortest path is a fuzzy interval with a decreasing linear membership function. By the max–min criterion suggested in [R.E. Bellman, L.A. Zade, Decision-making in a fuzzy environment, Management Science 17B (1970) 141–164], the fuzzy shortest path problem can be treated as a mixed integer nonlinear programming problem. We show that this problem can be simplified into a bi-level programming problem that is very solvable. Here, we propose an efficient algorithm, based on the parametric shortest path problem for solving the bi-level programming problem. An illustrative example is given to demonstrate our proposed algorithm.     

Vehicle routing problems with alternative paths: An application to on-demand transportation

The class of vehicle routing problems involves the optimization of freight or passenger transportation activities. These problems are generally treated via the representation of the road network as a weighted complete graph. Each arc of the graph represents the shortest route for a possible origin–destination connection. Several attributes can be defined for one arc (travel time, travel cost, etc.), but the shortest route modeled by this arc is computed according to a single criterion, generally travel time. Consequently, some alternative routes proposing a different compromise between the attributes of the arcs are discarded from the solution space. We propose to consider these alternative routes and to evaluate their impact on solution algorithms and solution values through a multigraph representation of the road network. We point out the difficulties brought by this representation for general vehicle routing problems, which drives us to introduce the so-called fixed sequence arc selection problem (FSASP). We propose a dynamic programming solution method for this problem. In the context of an on-demand transportation (ODT) problem, we then propose a simple insertion algorithm based on iterative FSASP solving and a branch-and-price exact method. Computational experiments on modified instances from the literature and on realistic data issued from an ODT system in the French Doubs Central area underline the cost savings brought by the proposed methods using the multigraph model.     

A re-scaled twin augmented Lagrangian algorithm for saddle point seeking

In [A. Ouorou, A primal-dual algorithm for monotropic programming and its application to network optimization, Computational Optimization and Application 15 (2002) 125–143], a block-wise Gauss–Seidel method has been developed for monotropic programming problems, using two different quadratic augmented Lagrangian functions defined for the primal and the dual problems. In this paper, we extend the concept by introducing a nonlinear re-scaling principle obtained recently by Polyak [R. Polyak, Nonlinear rescaling vs smoothing technique in constrained optimization, Mathematical Programming 92 (2002) 197–235].     

Interactive TOPSIS algorithms for solving multi-level non-linear multi-objective decision-making problems

This paper extended the concept of the technique for order preference by similarity to ideal solution (TOPSIS) to develop a methodology for solving multi-level non-linear multi-objective decision-making (MLN-MODM) problems of maximization-type. Also, two new interactive algorithms are presented for the proposed TOPSIS approach for solving these types of mathematical programming problems. The first proposed interactive TOPSIS algorithm includes the membership functions of the decision variables for each level except the lower level of the multi-level problem. These satisfactory decisions are evaluated separately by solving the corresponding single-level MODM problems. The second proposed interactive TOPSIS algorithm lexicographically solves the MODM problems of the MLN-MOLP problem by taking into consideration the decisions of the MODM problems for the upper levels. To demonstrate the proposed algorithms, a numerical example is solved and compared the solutions of proposed algorithms with the solution of the interactive algorithm of Osman et al. (2003) [4]. Also, an example of an application is presented to clarify the applicability of the proposed TOPSIS algorithms in solving real world multi-level multi-objective decision-making problems.     

Model and algorithm for inventory/routing decision in a three-echelon logistics system

This paper addresses an integrated inventory and routing problem in a three-echelon logistics system, which consists of a supplier, a central warehouse and a group of retailers. The inventory decision of each member and the routing decision among members of the system are made simultaneously, with the objective of minimizing the overall average cost of the system. A strategy named fixed partition and power-of-two (FP–POT) is proposed for the considered problem and a variable large neighborhood search (VLNS) algorithm, which is a special case of variable neighborhood search (VNS) algorithm, is developed. The efficiency of the strategy as well as the algorithm is illustrated by comparing computational results with a lower bound. The advantage of the proposed VLNS algorithm is further shown by getting better results for the problems in a two-echelon logistics system, which have been solved by a Tabu Search algorithm recently.     

The inverse Fermat–Weber problem

Given n points in the plane with nonnegative weights, the inverse Fermat–Weber problem consists in changing the weights at minimum cost such that a prespecified point in the plane becomes the Euclidean 1-median. The cost is proportional to the increase or decrease of the corresponding weight. In case that the prespecified point does not coincide with one of the given n points, the inverse Fermat–Weber problem can be formulated as linear program. We derive a purely combinatorial algorithm which solves the inverse Fermat–Weber problem with unit cost using O(n) greedy-like iterations where each of them can be done in constant time if the points are sorted according to their slopes. If the prespecified point coincides with one of the given n points, it is shown that the corresponding inverse problem can be written as convex problem and hence is solvable in polynomial time to any fixed precision.     

A new method for solving fully fuzzy linear programming problems

Lotfi et al. [Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution, Appl. Math. Modell. 33 (2009) 3151–3156] pointed out that there is no method in literature for finding the fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems and proposed a new method to find the fuzzy optimal solution of FFLP problems with equality constraints. In this paper, a new method is proposed to find the fuzzy optimal solution of same type of fuzzy linear programming problems. It is easy to apply the proposed method compare to the existing method for solving the FFLP problems with equality constraints occurring in real life situations. To illustrate the proposed method numerical examples are solved and the obtained results are discussed.     

Converging Marriage in Honey-Bees Optimization and Application to Stochastic Dynamic Programming

In this paper, we first refine a recently proposed metaheuristic called “Marriage in Honey-Bees Optimization” (MBO) for solving combinatorial optimization problems with some modifications to formally show that MBO converges to the global optimum value. We then adapt MBO into an algorithm called “Honey-Bees Policy Iteration” (HBPI) for solving infinite horizon-discounted cost stochastic dynamic programming problems and show that HBPI also converges to the optimal value.