Developing work schedules for an inter-city transit system with multiple driver types and fleet types

In developing work schedules, the job assignment flexibility exploits the variety of available skills, thus enabling the assignment of workers to perform different jobs. In this study, we investigate the problem of finding the mix of primary and secondary jobs in short term work schedules to meet, at minimum cost, the daily service requirements of an inter-city bus transit firm in Andra Pradesh India operating multiple fleet types. We formulate the problem as a set covering model with resource allocation constraints. We develop a branch-and-price procedure to solve the model. Computational results are provided.     

The splitting of variables and constraints in the formulation of integer programming models

The splitting of variables in an integer programming model into the sum of other variables can allow the constraints to be disaggregated, leading to a more constrained (tighter) linear programming relaxation. Well known examples of such reformulations are quoted from the literature. They can be viewed as instances of some general methods of performing such reformulations, namely disjunctive formulations, partial network reformulations and a method based on the introduction of auxiliary variables.     

The transportation problem with exclusionary side constraints and two branch-and-bound algorithms

The transportation problem with exclusionary side constraints, a practical distribution and logistics problem, is formulated as a 0–1 mixed integer programming model. Two branch-and-bound (B&B) algorithms are developed and implemented in this study to solve this problem. Both algorithms use the Driebeek penalties to strengthen the lower bounds so as to fathom some of the subproblems, to peg variables, and to guide the selection of separation variables. One algorithm also strongly exploits the problem structure in selecting separation variables in order to find feasible solutions sooner. To take advantage of the underlying network structure of the problem, the algorithms employ the primal network simplex method to solve network relaxations of the problem. A computational experiment was conducted to test the performance of the algorithms and to characterize the problem difficulty. The commercial mixed integer programming software CPLEX and an existing special purpose algorithm specifically designed for this problem were used as benchmarks to measure the performance of the algorithms. Computational results show that the new algorithms completely dominate the existing special purpose algorithm and run from two to three orders of magnitude faster than CPLEX.     

A o(n logn) algorithm for LP knapsacks with GUB constraints

A specialization of the dual simplex method is developed for solving the linear programming (LP) knapsack problem subject to generalized upper bound (GUB) constraints. The LP/GUB knapsack problem is of interest both for solving more general LP problems by the dual simplex method, and for applying surrogate constraint strategies to the solution of 0–1 Multiple Choice integer programming problems. We provide computational bounds for our method of o(n logn), wheren is the total number of problem variables. These bounds reduce the previous best estimate of the order of complexity of the LP/GUB knapsack problem (due to Witzgall) and provide connections to computational bounds for the ordinary knapsack problem.We further provide a variant of our method which has only slightly inferior worst case bounds, yet which is ideally suited to solving integer multiple choice problems due to its ability to post-optimize while retaining variables otherwise weeded out by convex dominance criteria.     

Zero-one integer programs with few constraints —Efficient branch and bound algorithms

The 0–1 integer programming problem and its special case, the 0–1 knapsack problem are frequently encountered in modeling various design and decision making processes. This paper is a follow-up paper to [4] and deals with the development of an effective solution procedure for 0–1 integer programs with few constraints. Detailed computational experiments are carried out and different separation, branching and bounding rules are compared using an experimental branch and bound code. An efficient branch and bound procedure is developed, tested and compared with previously developed optimal algorithms. It is suggested that this procedure may also be used as a heuristic method for large problems by early termination of the tree search. This scheme is tested and found to be very effective.     

Optimizing glass coating lines: MIP model and valid inequalities

Glass coating is a specific transformation aiming at improving glass performance. The work presented in this paper deals with the determination of the optimal configuration of the production lines used to perform this operation. We propose a first MIP formulation of the problem and then discuss several types of valid inequalities for improving it. The main idea is to exploit explicit or implicit binary exclusion constraints to derive stronger valid inequalities: the maximal clique constraints. Efficient (polynomial time) separation algorithms exploiting special structure of the problem are described, giving rise to a cutting-plane generation procedure for strengthening the initial formulation. The computational study carried out shows that, with the enhanced formulation, good solutions can be obtained within reasonable computation times using currently available integer programming software.     

Transportation problem with nonlinear side constraints a branch and bound approach

In a container terminal management, we are often confronted with the following problem: how to assign a reasonable depositing position for an arriving container, so that the efficiency of searching for and loading of a container later can be increased. In this paper, the problem is modeled as a transportation problem with nonlinear side constraints (TPNSC). The reason of nonlinear side constraints arising is that some kinds of containers cannot be stacked in the same row (the space of storage yard is properly divided into several rows). A branch and bound algorithm is designed to solve this problem. The algorithm is based on the idea of using disjunctive arcs (branches) for resolving conflicts that are created whenever some conflicting kinds of containers are deposited in the same row. During the branch and bound, the candidate problems are transformed into classical transportation problems, so that the efficient transportation algorithm can be applied, at the same time the reoptimization technique is employed during the branch and bound. Further, we design a heuristic to obtain a feasible initial solution for TPNSC in order to prune some candidates as early and/or as much as possible. We report computational results on randomly generated problems.     

The ring-star problem: A new integer programming formulation and a branch-and-cut algorithm

A ring star in a graph is a subgraph that can be decomposed into a cycle (or ring) and a set of edges with exactly one vertex in the cycle. In the minimum ring-star problem (mrsp) the cost of a ring star is given by the sum of the costs of its edges, which vary, depending on whether the edge is part of the ring or not. The goal is to find a ring-star spanning subgraph minimizing the sum of all ring and assignment costs. In this paper we show that the mrsp can be reduced to a minimum (constrained) Steiner arborescence problem on a layered graph. This reduction is used to introduce a new integer programming formulation for the mrsp. We prove that the dual bound generated by the linear relaxation of this formulation always dominates the one provided by an early model from the literature. Based on our new formulation, we developed a branch-and-cut algorithm for the mrsp. On the primal side, we devised a grasp heuristic to generate good upper bounds for the problem. Computational tests with these algorithms were conducted on a benchmark of public domain. In these experiments both our exact and heuristics algorithms had excellent performances, noticeably in dealing with instances whose optimal solution has few vertices in the ring. In addition, we also investigate the minimum spanning caterpillar problem (mscp) which has the same input as the mrsp and admits feasible solutions that can be viewed as ring stars with paths in the place of rings. We present an easy reduction of the mscp to the mrsp, which makes it possible to solve to optimality instances of the former problem too. Experiments carried out with the mscp revealed that our branch-and-cut algorithm is capable to solve to optimality instances with up to 200 vertices in reasonable time.     

Evaluating the strategic behavior of cement producers: An equilibrium problem with equilibrium constraints

This paper investigates the equilibria reached by a number of strategic producers in the cement sector through a technological representation of the market. We present a bilevel model for each producer that characterizes its profit maximizing behavior. In the bilevel model, the upper-level problem of each producer is constrained by a lower-level market clearing problem representing cement trading and whose objective function corresponds to social welfare. Replacing the lower level problem by its optimality condition yields a Mathematical Program with Equilibrium Constraints (MPEC). Then, all strategic producers are jointly considered. Representing their interaction requires solving jointly the interrelated MPECs of all producers, which results in an Equilibrium Problem with Equilibrium Constraints (EPEC).A parametric analysis concerning cost, capacity and demand fluctuations has been conducted. Our analysis shows that the European cement sector is mature and has lost its competitiveness; African cement market can assume a prominent role in international markets in the coming future if investments in new and efficient capacity are carried out. Finally, the Far East will remain the reference exporter of cement at worldwide level.     

Integer matrices with constraints on leading partial row and column sums

Consider the problem of finding an integer matrix that satisfies given constraints on its leading partial row and column sums. For the case in which the specified constraints are merely bounds on each such sum, an integer linear programming formulation is shown to have a totally unimodular constraint matrix. This proves the polynomial-time solvability of this case. In another version of the problem, one seeks a zero-one matrix with prescribed row and column sums, subject to certain near-equality constraints, namely, that all leading partial row (respectively, column) sums up through a given column (respectively, row) are within unity of each other. This case admits a polynomial reduction to the preceding case, and an equivalent reformulation as a maximum-flow problem. The results are developed in a context that relates these two problems to consistent matrix rounding.     

Product line pricing for services with capacity constraints and dynamic substitution

In this paper, we address a service provider’s product line pricing problem for substitutable products in services, such as concerts, sporting events, or online advertisements. For each product, a static price is selected from a pre-defined set such that the total revenue is maximised. The products are differentiated by some of their attributes, and their availability is restricted due to individual capacity constraints. Furthermore, they are simultaneously sold during a common selling period at the end of which the service is delivered. Consumers differ from one another with respect to their willingness to pay, and, hence, their reservation prices vary depending on the product. In the event of a purchase, they choose the product that maximises their consumer surplus.     

A MIP approach for some practical packing problems: Balancing constraints and tetris-like items

This paper considers packing problems with balancing conditions and items consisting of clusters of parallelepipeds (mutually orthogonal, i.e. tetris-like items). This issue is quite frequent in space engineering and a real-world application deals with the Automated Transfer Vehicle project (funded by the European Space Agency), at present under development. A Mixed Integer Programming (MIP) approach is proposed. The three-dimensional single bin packing problem is considered. It consists of orthogonally placing, with possibility of rotation, the maximum number of parallelepipeds into a given parallelepiped. A MIP formulation of the problem is reported together with a MIP-based heuristic approach. Balancing conditions are furthermore examined, as well as the orthogonal placement (with rotation) of tetris-like items into a rectangular domain.Received: September 2003, Revised: February 2004, AMS classification: 90B99, 05B40, 90C90, 90C59Thanks are due to T. A. Ciriani for the important suggestions given for the whole paper and to S. Gliozzi (IBM, Business Consulting Services) for the significant support offered, in particular in discussing the topics presented in Sect. 2.1.     

Improved bounds for the traveling umpire problem: A stronger formulation and a relax-and-fix heuristic

Given a double round-robin tournament, the traveling umpire problem (TUP) consists of determining which games will be handled by each one of several umpire crews during the tournament. The objective is to minimize the total distance traveled by the umpires, while respecting constraints that include visiting every team at home, and not seeing a team or venue too often. We strengthen a known integer programming formulation for the TUP and use it to implement a relax-and-fix heuristic that improves the quality of 24 out of 25 best-known feasible solutions to instances in the TUP benchmark. We also improve all best-known lower bounds for those instances and, for the first time, provide lower bounds for instances with more than 16 teams.     

A mixed integer linear programming approach to minimize the number of late jobs with and without machine availability constraints

This study investigates scheduling problems that occur when the weighted number of late jobs that are subject to deterministic machine availability constraints have to be minimized. These problems can be modeled as a more general job selection problem. Cases with resumable, non-resumable, and semi-resumable jobs as well as cases without availability constraints are investigated. The proposed efficient mixed integer linear programming approach includes possible improvements to the model, notably specialized lifted knapsack cover cuts. The method proves to be competitive compared with existing dedicated methods: numerical experiments on randomly generated instances show that all 350-job instances of the test bed are closed for the well-known problem 1|_ri|∑_wiUi$1|r_i|\sum w_i{U}_i$. For all investigated problem types, 98.4% of 500-job instances can be solved to optimality within 1 hour.     

Parallel machine scheduling with additional resources: Notation,classification, models and solution methods

Majority of parallel machine scheduling studies consider machine as the only resource. However, in most real-life manufacturing environments, jobs may require additional resources, such as automated guided vehicles, machine operators, tools, pallets, dies, and industrial robots, for their handling and processing. This paper presents a review and discussion of studies on the parallel machine scheduling problems with additional resources. Papers are surveyed in five main categories: machine environment, additional resource, objective functions, complexity results and solution methods, and other important issues. The strengths and weaknesses of the literature together with open areas for future studies are also emphasized. Finally, extensions of integer programming models for two main classes of related problems are given and conclusions are drawn based on computational studies.     

Freight transportation in railway networks with automated terminals: A mathematical model and MIP heuristic approaches

In this paper we propose a planning procedure for serving freight transportation requests in a railway network with fast transfer equipment at terminals. We consider a transportation system where different customers make their requests (orders) for moving boxes, i.e., either containers or swap bodies, between different origins and destinations, with specific requirements on delivery times. The decisions to be taken concern the route (and the corresponding sequence of trains) that each box follows in the network and the assignment of boxes to train wagons, taking into account that boxes can change more than one train and that train timetables are fixed.The planning procedure includes a pre-analysis step to determine all the possible sequences of trains for serving each order, followed by the solution of a 0-1 linear programming problem to find the optimal assignment of each box to a train sequence and to a specific wagon for each train in the sequence. This latter is a generalized assignment problem which is NP-hard. Hence, in order to find good solutions in acceptable computation times, two MIP heuristic approaches are proposed and tested through an experimental analysis considering realistic problem instances.     

Branch and bound algorithm for a transfer line design problem: Stations with sequentially activated multi-spindle heads

In this paper, a new transfer line balancing problem is considered. The main difference from the simple assembly line balancing problem is that the operations are grouped into blocks (batches). All the operations of the same block are carried out simultaneously by one piece of equipment (multi-spindle head). Nevertheless, the blocks assigned to the same workstation are executed in series. The line cost consists of the sum of block and workstation costs. At the considered line design stage, the set of all available blocks is given. The aim is to find a subset from the given set of available blocks and to find a partition of this subset to workstations such that each operation is executed once and the line cost is minimal while all the precedence, cycle time, and compatibility (operation inclusion and block exclusion) constraints are respected. A new lower bound based on solving a special set partitioning problem is presented and a branch and bound algorithm is developed. The experimental results prove the quality of the lower bound and applicability of the suggested branch and bound algorithm for medium sized industrial cases.     

Solution methods for the bi-objective (cost-coverage) unconstrained facility location problem with an illustrative example

The Colombian coffee supply network, managed by the Federación Nacional de Cafeteros de Colombia (Colombian National Coffee-Growers Federation), requires slimming down operational costs while continuing to provide a high level of service in terms of coverage to its affiliated coffee growers. We model this problem as a biobjective (cost-coverage) uncapacitated facility location problem (BOUFLP). We designed and implemented three different algorithms for the BOUFLP that are able to obtain a good approximation of the Pareto frontier. We designed an algorithm based on the Nondominated Sorting Genetic Algorithm; an algorithm based on the Pareto Archive Evolution Strategy; and an algorithm based on mathematical programming. We developed a random problem generator for testing and comparison using as reference the Colombian coffee supply network with 29 depots and 47 purchasing centers. We compared the algorithms based on the quality of the approximation to the Pareto frontier using a nondominated space metric inspired on Zitzler and Thiele's. We used the mathematical programming-based algorithm to identify unique tradeoff opportunities for the reconfiguration of the Colombian coffee supply network. Finally, we illustrate an extension of the mathematical programming-based algorithm to perform scenario analysis for a set of uncapacitated location problems found in the literature.     

Developing new formulations and relaxations of single stage capacitated warehouse location problem (SSCWLP): Empirical investigation for assessing relative strengths and computational effort

In literature, single/multistage warehouse location problems have been attempted by Geoffrion and Graves [A.M. Geoffrion, G.W. Graves, Multicommodity distribution system design by Benders decomposition, Management Science 2 (1974) 82–114] and Sharma [R.R.K. Sharma, Modeling a fertilizer distribution system, European Journal of Operational Research 51 (1991) 24–34] among others and they have given completely different formulations. We use the formulation style given by Sharma and Sharma [R.R.K. Sharma, K.D. Sharma, A new dual based procedure for the transportation problem, European Journal of Operational Research 122 (3) (2000) 96–109] to develop variety of constraints that link real and 0–1 integer variables; thus developing many formulations of single stage capacitated warehouse location problem (SSCWLP). We relax the integer constraints on 0–1 variables to obtain their relaxations.     

Mathematical programs with vanishing constraints: constraint qualifications,their applications,and a new regularization method

ABSTRACT

We propose a new family of relaxation schemes for mathematical programs with vanishing constraints that extend the relaxation of Hoheisel, Kanzow & Schwartz from 2012. We discuss the properties of the sequence of relaxed non-linear programs as well as stationary properties of limiting points. Our relaxation schemes have the desired property of converging to an M-stationary point. A main advantage of this new method is to converge to an S-stationary point satisfying MPVC-LICQ for a large class of problem. We also study MPVC constraint qualification connected to this study and prove convergence of the method under the new MPVC-CRSC. Additionally, we obtain the new MPVC-wGCQ and prove that it is the weakest MPVC constraint qualification.