Economic inefficiency measurement of input spending when decision-making units face different input prices

In a recent paper, Kaoru Tone (J Opl Res Soc (2002) 2: 429–444) showed that when the Farrell measure of cost efficiency is estimated for two firms that have different input prices, a firm with higher costs can be deemed more efficient than a firm with lower costs. As an alternative approach, Tone proposed a radial cost efficiency measure that is estimated using levels of spending on each input, rather than input quantities. Thus, firms with higher costs are less efficient than firms with lower costs. In this paper, we extend Tone's approach by allowing for non-radial changes in spending. Our approach builds on earlier work by Luenberger (J Math Econ (1992) 21: 461–481) and Chambers et al (J Econ Theo (1996) 70: 407–419) who use directional distance functions to measure inefficiency. We provide an example and illustration of our approach using Japanese bank data.     

Integer programming to schedule a hierarchical workforce with variable demands

We consider a hierarchical workforce in which a higher qualified worker can substitute for a lower qualified one, but not vice versa. Daily labor requirements within a week may vary, but each worker must receive n off-days in the week. This problem has been considered by Hung (R. Hung, Eur. J. Oper. Res. 78(1) (1994) 49–57), who discusses a necessary and sufficient condition for a labor mix to be feasible and presents a simple one-pass method that frequently gives the least cost labor mix. We show in this paper that the integer programming approach is well suited for solving this problem: the definition of the integer programming model is simple, its implementation is immediate by using, for example, the Mathematical programming language (MPL) and the integer programming solver XA, the computation times are low (generally a few seconds on a small microcomputer) and finally the powerful of the integer programming approach allows us to extend the model in two interesting directions.     

A note on “branch-and-price approach for the multi-skill project scheduling problem”

In a recent paper published in Optimization Letters, Montoya et al. (Optim Lett 8:1721–1734, 2014) proposed a branch-and-price approach for a multi-skill project scheduling problem. In that paper, an integer linear programming formulation was first introduced which, unfortunately, has a number of inconsistences. At the interest of mathematical rigor, in this note, we refine such formulation.     

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.     

Design of manufacturing cells in the presence of alternative cell locations and material transporters

This paper addresses two significant issues in the design of cellular manufacturing (CM) systems: (i) the availability of alternative locations for a cell, and (ii) the use of alternative routes to move part loads between cells when the capacity of the material transporter (MT) employed is limited. In addition, several other important factors in the design of CM systems including machine capacity limitations, batches of part demands, non-consecutive operations of parts, and maximum number of machines assigned to a cell are considered. A nonlinear programming model, comprised of binary and general integer variables, is formulated for the research problem. A higher-level heuristic solution algorithm based upon a concept known as ‘tabu search’ is presented for solving industry-size problems. Six different versions of the heuristic are developed to investigate the impact of long-term memory and the use of fixed versus variable tabu-list sizes. Explicit method-based techniques are developed to convert the original nonlinear programming model into an equivalent mixed (binary)-integer linear programming model in order to test the efficacy of the proposed solution technique for solving small problem instances. The solutions obtained from the heuristics have average deviation of less than 3% of the optimal solutions, and require less than a minute in comparison with optimizing methods that needed 1–10 h of computation time. A carefully designed statistical experiment is used to compare the performance of the heuristics by solving three different problem structures, ranging from four to 30 parts, and three to nine locations. The experiment shows that as the problem size increases, the tabu-search-based heuristic using fixed tabu list size and long-term memory based on minimal frequency strategy is preferred over the other heuristics.     

A mathematical programming approach to optimise insurance premium pricing within a data mining framework

In this paper we provide evidence of the benefits of an approach which combines data mining and mathematical programming to determining the premium to charge automobile insurance policy holders in order to arrive at an optimal portfolio. An non-linear integer programming formulation is proposed to determine optimal premiums based on the insurer's need to find a balance between profitability and market share. The non-linear integer programming approach to solving this problem is used within a data mining framework which consists of three components: classifying policy holders into homogenous risk groups and predicting the claim cost of each group using k-means clustering; determining the price sensitivity (propensity to pay) of each group using neural networks; and combining the results of the first two components to determine the optimal premium to charge. We have earlier presented the results of the first two components. In this paper we present the results of the third component. Using our approach, we have been able to increase revenue without affecting termination rates and market share.     

A dynamic programming algorithm for scheduling jobs in a two-machine open shop with an availability constraint

This paper studies a two-machine open shop scheduling problem with an availability constraint, ie we assume that a machine is not always available and that the processing of the interrupted job can be resumed when the machine becomes available again. We consider the makespan minimization as criterion. This problem is NP-hard. We develop a pseudo-polynomial time dynamic programming algorithm to solve the problem optimally when the machine is not available at time s>0. Then, we propose a mixed integer linear programming formulation, that allows to solve instances with up to 500 jobs optimally in less than 5?min with CPLEX solver. Finally, we show that any heuristic algorithm has a worst-case error bound of 1.     

A branch-and-cut algorithm for the capacitated open vehicle routing problem

In open vehicle routing problems, the vehicles are not required to return to the depot after completing service. In this paper, we present the first exact optimization algorithm for the open version of the well-known capacitated vehicle routing problem (CVRP). The algorithm is based on branch-and-cut. We show that, even though the open CVRP initially looks like a minor variation of the standard CVRP, the integer programming formulation and cutting planes need to be modified in subtle ways. Computational results are given for several standard test instances, which enables us for the first time to assess the quality of existing heuristic methods, and to compare the relative difficulty of open and closed versions of the same problem.     

All Colors Shortest Path problem on trees

Given an edge weighted tree T(V, E), rooted at a designated base vertex \(r \in V\), and a color from a set of colors \(C=\{1,\ldots ,k\}\) assigned to every vertex \(v \in V\), All Colors Shortest Path problem on trees (ACSP-t) seeks the shortest, possibly non-simple, path starting from r in T such that at least one node from every distinct color in C is visited. We show that ACSP-t is NP-hard, and also prove that it does not have a constant factor approximation. We give an integer linear programming formulation of ACSP-t. Based on a linear programming relaxation of this formulation, an iterative rounding heuristic is proposed. The paper also explores genetic algorithm and tabu search to develop alternative heuristic solutions for ACSP-t. The performance of all the proposed heuristics are evaluated experimentally for a wide range of trees that are generated parametrically.     

Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers

The Quadratic Assignment Problem (QAP) can be solved by linearization, where one formulates the QAP as a mixed integer linear programming (MILP) problem. On the one hand, most of these linearizations are tight, but rarely exploited within a reasonable computing time because of their size. On the other hand, Kaufman and Broeckx formulation (Eur. J. Oper. Res. 2(3):204–211, 1978) is the smallest of these linearizations, but very weak. In this paper, we analyze how the Kaufman and Broeckx formulation can be tightened to obtain better QAP-MILP formulations. As shown in our numerical experiments, these tightened formulations remain small but computationally effective to solve the QAP by means of general purpose MILP solvers.     

DEA based dimensionality reduction for classification problems satisfying strict non-satiety assumption

This study shows how data envelopment analysis (DEA) can be used to reduce vertical dimensionality of certain data mining databases. The study illustrates basic concepts using a real-world graduate admissions decision task. It is well known that cost sensitive mixed integer programming (MIP) problems are NP-complete. This study shows that heuristic solutions for cost sensitive classification problems can be obtained by solving a simple goal programming problem by that reduces the vertical dimension of the original learning dataset. Using simulated datasets and a misclassification cost performance metric, the performance of proposed goal programming heuristic is compared with the extended DEA-discriminant analysis MIP approach. The holdout sample results of our experiments shows that the proposed heuristic approach outperforms the extended DEA-discriminant analysis MIP approach.     

Parametric Integer Programming Analysis: A Contraction Approach

An algorithm is presented for solving families of integer linear programming problems in which the problems are "related" by having identical objective coefficients and constraint matrix coefficients. The righthand-side constants have the form b + θd where b and d are conformable vectors and θ varies from zero to one.The approach consists primarily of solving the most relaxed problem (θ = 1) using cutting planes and then contracting the region of feasible integer solutions in such a manner that the current optimal integer solution is eliminated.The algorithm was applied to 1800 integer linear programming problems with reasonable success. Integer programming problems which have proved to be unsolvable using cutting planes have been solved by expanding the region of feasible integer solutions (θ = 1) and then contracting to the original region.     

Optimizing a multi-stage production/inventory system by DC programming based approaches

This paper deals with optimizing the cost of set up, transportation and inventory of a multi-stage production system in presence of bottleneck. The considered optimization model is a mixed integer nonlinear program. We propose two methods based on DC (Difference of Convex) programming and DCA (DC Algorithm)—an innovative approach in nonconvex programming framework. The mixed integer nonlinear problem is first reformulated as a DC program and then DCA is developed to solve the resulting problem. In order to globally solve the problem, we combine DCA with a Branch and Bound algorithm (BB-DCA). A convex minorant of the objective function is introduced. DCA is used to compute upper bounds while lower bounds are calculated from a convex relaxation problem. The numerical results compared with those of COUENNE (http://www.coin-or.org/download/binary/Couenne/), a solver for mixed integer nonconvex programming, show the rapidity and the ?-globality of DCA in almost cases, as well as the efficiency of the combined DCA-Branch and Bound algorithm. We also propose a simple heuristic algorithm which is proved by experimental results to be better than an existing heuristic in the literature for this problem.     

Mond–Weir type second order symmetric duality in non-differentiable minimax mixed integer programming problems

A pair of Mond–Weir type non-differentiable second order symmetric minimax mixed integer primal and dual problems in mathematical programming is formulated. Symmetric and self-duality theorems are then established under second order F-pseudo-convexity assumptions. Several known results including that of Gulati and Ahmad [Eur. J. Oper. Res. 101 (1997) 122], Hou and Yang [J. Math. Anal. Appl. 255 (2001) 491] and Mond and Schechter [Bull. Aust. Math. Soc. 53 (1996) 177], as well as others are obtained as special cases.     

Fair transfer price and inventory holding policies in two-enterprise supply chains

A key issue in supply chain optimisation involving multiple enterprises is the determination of policies that optimise the performance of the supply chain as a whole while ensuring adequate rewards for each participant.In this paper, we present a mathematical programming formulation for fair, optimised profit distribution between echelons in a general multi-enterprise supply chain. The proposed formulation is based on an approach applying the Nash bargaining solution for finding optimal multi-partner profit levels subject to given minimum echelon profit requirements.The overall problem is first formulated as a mixed integer non-linear programming (MINLP) model. A spatial and binary variable branch-and-bound algorithm is then applied to the above problem based on exact and approximate linearisations of the bilinear terms involved in the model, while at each node of the search tree, a mixed integer linear programming (MILP) problem is solved. The solution comprises inter-firm transfer prices, production and inventory levels, flows of products between echelons, and sales profiles.The applicability of the proposed approach is demonstrated by a number of illustrative examples based on industrial processes.     

Branch-and-price for <Emphasis Type="Italic">p</Emphasis>-cluster editing

Given an input graph, the p-cluster editing problem consists of minimizing the number of editions, i.e., additions and/or deletions of edges, so as to create p vertex-disjoint cliques (clusters). In order to solve this \({\mathscr {NP}}\)-hard problem, we propose a branch-and-price algorithm over a set partitioning based formulation with exponential number of variables. We show that this formulation theoretically dominates the best known formulation for the problem. Moreover, we compare the performance of three mathematical formulations for the pricing subproblem, which is strongly \({\mathscr {NP}}\)-hard. A heuristic algorithm is also proposed to speedup the column generation procedure. We report improved bounds for benchmark instances available in the literature.     

An Improved Formulation for the Job-Shop Scheduling Problem

This paper presents an extension of an earlier integer programming model developed by other authors to formulate a general n-job, m-machine job-shop problem. The new formulation involves substantially fewer functional constraints at the expense of an increase in the number of upper bound variables. This reduction of functional constraints, together with the imposition of upper and lower bounds on the objective value, significantly reduces the computation time for solving the integer model for the job-shop scheduling problem.     

Solving a capacitated hub location problem

In this paper we address a problem consisting of determining the routes and the hubs to be used in order to send, at minimum cost, a set of commodities from sources to destinations in a given capacitated network. The capacities and costs of the arcs and hubs are given, and the arcs connecting the hubs are not assumed to create a complete graph. We present a mixed integer linear programming formulation and describe two branch-and-cut algorithms based on decomposition techniques. We evaluate and compare these algorithms on instances with up to 25 commodities and 10 potential hubs. One of the contributions of this paper is to show that a Double Benders’ Decomposition approach outperforms the standard Benders’ Decomposition, which has been widely used in recent articles on similar problems. For larger instances we propose a heuristic approach based on a linear programming relaxation of the mixed integer model. The heuristic turns out to be very effective and the results of our computational experiments show that near-optimal solutions can be derived rapidly.     

A unified solving approach for two and three dimensional coverage problems in sensor networks

The problem of designing a wired or a wireless sensor network to cover, monitor and/or control a region of interest has been widely treated in literature. This problem is referred to in literature as the sensor placement problem (SPP) and in the most general case it consists in determining the number and the location of one or more kind of sensors with the aim of covering all the region of interest or a significant part of it. In this paper we propose a unified and stepwise solving approach for two and three dimensional coverage problems to be used in omni-directional and directional sensor networks. The proposed approach is based on schematizing the region of interest and the sensor potential locations by a grid of points and representing the sensor coverage area by a circle or by a circular sector. On this basis, the SPP is reduced to an optimal coverage problem and can be formulated by integer linear programming (ILP) models. We will resume the main ILP models used in our approach, highlighting, for each of them, the specific target to be achieved and the design constraints taken into account. The paper concludes with an application of the proposed approach to a real test case and a discussion of the obtained results.     

A MIP-based slicing heuristic for three-dimensional bin packing

The mixed-case palletization problem is a common problem in warehousing and logistics where boxes of rectangular shapes are stacked on top of each other to form pallets. The problem shares common features with three-dimensional bin packing but requires boxes to be adequately supported. We propose a mixed integer programming formulation that maximizes the density of the bottom layers and the compactness of the pallet to ensure stability for top layers. We use a relative-position formulation with slicing that minimizes height, maximizes the fill rate of slices, and pushes boxes towards the vertical axis in order to consolidate fragmented space. Apart from common non-overlap and dimension-related constraints, we explicitly model the fill rates and force lower slices to have an equal or higher density than upper slices. As expected, the formulation could only handle small instances. To tackle larger instances, we embedded the formulation in an iterative approach that packs subsets of boxes sequentially. The approach was found to provide stable pallets and to outperform the branch-and-bound approach of Martello et al. (Oper Res 48(2):256–267, 2000).