IP Solver Research Articles

Local Search-Based Metaheuristic Methods for the Solid Waste Collection Problem

The solid waste collection problem refers to truck route optimisation to collect waste from containers across various locations. Recent concerns exist over the impact of solid waste management on the environment. Hence, it is necessary to find feasible routes while minimising operational costs and fuel consumption. In this paper, in order to reduce fuel consumption, the number of trucks used is considered in the objective function along with the waste load and the travelling time. With the current computational capabilities, finding an optimal solution is challenging. Thus, this study aims to investigate the effect of well-known metaheuristic methods on this problem’s objective function and computational times. The routing solver in the Google OR-tools solver is utilised with three well-known metaheuristic methods for neighbourhood exploration: a guided local search (GLS), a tabu search (TS), and simulated annealing (SA), with two initialisation strategies, Clarke and Wright’s algorithm and the nearest neighbour algorithm. Results showed that optimal solutions are found in faster computational times than using only an IP solver, especially for large instances. Local search methods, notably GLS, have significantly improved the route construction process. The nearest neighbour algorithm has often outperformed the Clarke and Wright's methods. The findings here can be applied to improve operations in Saudi Arabia’s waste management sector.

Coping with shortages caused by disruptive events in automobile supply chains

AbstractUnpredictable disruptive events significantly increase the difficulty of the management of automobile supply chains. In this paper, we propose an automobile production planning problem with component chips substitution in a finite planning horizon. The shortage of one chip can be compensated by another chip of the same type with a higher‐end feature at an additional cost. Therefore, the automobile manufacturer can divert the on‐hand inventory of chips to product lines that are more profitable in the event of shortages caused by supply chain disruptions. To cope with this, we propose a max‐min robust optimization model that captures the uncertain supplies of chips. We show that the robust model has a mixed‐integer programming equivalence that can be solved by a commercial IP solver directly. We compare the max‐min robust model with the corresponding deterministic and two‐stage stochastic models for the same problem through extensive numerical experiments. The computational results show that the max‐min robust model outperforms the other two models in terms of the average and worst‐case profits.

A new integer model for the management of booking stays at summer resorts

The integration of operational research techniques in the professional life has proved to be indispensable to accomplish delicate tasks in global companies, especially with the complications encountered in various disciplines. Such complications are mainly related to the links between physical and human resources in a management system at the operational level. In this paper, a new integer programming formulation of the booking stays problematic at summer resorts for a foundation members is proposed. The model was treated as a constrained assignment problem that provides many constraints related to the foundations’ rules and requirements. The objective of this model is to maximize a linear function and therefore satisfy the expressed choices of the majority of the foundation members concerning the period and residence in a particular center. The strength of our model lies in its ability to manage a large number of booking requests in the presence of very limited resources. This model is solvable by powerful existing software tools with IP solvers. Our new model was adopted by the Moroccan Mohammedia Foundation for Social Works of Judges and Justice Officials (MFSWJJO), and the obtained results were applied during the summer of 2018. The obtained results of experiments with real-data from MFSWJJO are reported.

Downward compatible loading optimization with inter-set cost in automobile outbound logistics

This paper addresses an essential loading optimization problem arising in the outbound logistics of automobiles. The problem is to assign a set of orders of finished cars to a set of heterogeneous auto-carriers and to deliver these finished cars from a depot to multiple destinations. The involved destinations are located within a relatively small distance to the depot. A value corresponding to each order represents its urgency level, and an inter-set cost is defined among destinations to estimate the related transportation cost. The objective of the problem is to maximize the weighted total value of the assigned orders minus the total inter-set cost subject to a full-load constraint and a downward compatible loading structure. An integer programming model and a column generation based algorithm are proposed. The proposed algorithm is implemented based on serial and parallel programming, respectively. Computational results based on randomly generated instances and a case study with real data show that the proposed algorithm can generate near-optimality solutions efficiently, and outperforms solving the integer programming model by an IP solver and a rule-based method applied in practice.

Plant propagation algorithm for nurse rostering

This paper investigates the nurse rostering problem (NRP), a challenging combinatorial optimisation problem that arises in healthcare institutions. We propose to solve it by using the plant propagation algorithm (PPA). As many successful metaheuristics, PPA is inspired by a life process. It emulates the strategy of reproduction and propagation of the strawberry plant. Before applying PPA, a variable-fixing procedure is used for heuristically discarding variables. In practice, it results in removing up to 99% of the variables without sacrificing solution quality. Elite solutions provided by PPA are used to further discard variables, leaving a very sparse NRP that can be solved directly by an IP solver. Computational and comparative results are presented on a widely used set of benchmark instances.

Exact approach to the electric bus fleet scheduling

Abstract In the recent years the research in the field of electromobility has been gaining a lot of interest. Research in the field of electromobility relates to various problems connected to the limited driving range of vehicles, locating of charging infrastructure and capacity of the electric network. These limitations should also be considered when solving the problem of transforming the public transport fleet of diesel engine busses into the fleet of electric vehicles. One of the tasks that should be solved is the problem of scheduling. This paper deals with the problem of assigning available electric buses to the set of service trips, in other words scheduling of electric buses. Electric buses have limited operation range and must be charged during the operation. Charging process needs more time than refueling of diesel busses and the number of charging points at charging station should be considered. In our case it is assumed that the charging is possible at the depot and at chosen locations. In the paper we propose a linear mathematical model that respects all necessary limitations. The performance of the model will be tested on the data set from the public transport system in the city of Žilina using a standard IP solver to evaluate the quality of the solution and computational time.

Scheduling a non-professional indoor football league: a tabu search based approach

This paper deals with a real-life scheduling problem of a non-professional indoor football league. The goal is to develop a schedule for a time-relaxed, double round-robin tournament which avoids close successions of games involving the same team in a limited period of time. This scheduling problem is interesting, because games are not planned in rounds. Instead, each team provides time slots in which they can play a home game, and time slots in which they cannot play at all. We present an integer programming formulation and a heuristic based on tabu search. The core component of this algorithm consists of solving a transportation problem, which schedules (or reschedules) all home games of a team. Our heuristic generates schedules with a quality comparable to those found with IP solvers, however with considerably less computational effort. These schedules were approved by the league organizers, and used in practice for the seasons 2009–2010 till 2016–2017.

A parallel approach to bi-objective integer programming

To obtain a better understanding of the trade-offs between various objectives, Bi-Objective Integer Programming (BOIP) algorithms calculate the set of all non-dominated vectors and present these as the solution to a BOIP problem. Historically, these algorithms have been compared in terms of the number of single-objective IPs solved and total CPU time taken to produce the solution to a problem. This is equitable, as researchers can often have access to widely differing amounts of computing power. However, the real world has recently seen a large uptake of multi-core processors in computers, laptops, tablets and even mobile phones. With this in mind, we look at how to best utilise parallel processing to improve the elapsed time of optimisation algorithms. We present two methods of parallelising the recursive algorithm presented by Ozlen, Burton and MacRae. Both new methods utilise two threads and improve running times. One of the new methods, the Meeting algorithm, halves running time to achieve near-perfect parallelisation. The results are compared with the efficiency of parallelisation within the commercial IP solver IBM ILOG CPLEX, and the new methods are both shown to perform better.

A hybrid integer and constraint programming approach to solve nurse rostering problems

The Nurse Rostering Problem can be defined as assigning a series of shift sequences (schedules) to several nurses over a planning horizon according to some limitations and preferences. The inherent benefits of generating higher-quality schedules are a reduction in outsourcing costs and an increase in job satisfaction of employees. In this paper, we present a hybrid algorithm, which combines Integer Programming and Constraint Programming to efficiently solve the highly-constrained Nurse Rostering Problem. We exploit the strength of IP in obtaining lower-bounds and finding an optimal solution with the capability of CP in finding feasible solutions in a co-operative manner. To improve the performance of the algorithm, and therefore, to obtain high-quality solutions as well as strong lower-bounds for a relatively short time, we apply some innovative ways to extract useful information such as the computational difficulty of instances and constraints to adaptively set the search parameters. We test our algorithm using two different datasets consisting of various problem instances, and report competitive results benchmarked with the state-of-the-art algorithms from the recent literature as well as standard IP and CP solvers, showing that the proposed algorithm is able to solve a wide variety of instances effectively.

Practical solutions for a dock assignment problem with trailer transportation

We study a distribution warehouse in which trailers need to be assigned to docks for loading or unloading. A parking lot is used as a buffer zone and transportation between the parking lot and the docks is performed by auxiliary resources called terminal tractors. Each incoming trailer has a known arrival time and each outgoing trailer a desired departure time. The primary objective is to produce a docking schedule such that the weighted sum of the number of late outgoing trailers and the tardiness of these trailers is minimized; the secondary objective is to minimize the weighted completion time of all trailers, both incoming and outgoing. The purpose of this paper is to produce high-quality solutions to large instances that are comparable to a real-life case. This will oblige us to abandon the guarantee of always finding an optimal solution, and we will instead look into a number of sub-optimal procedures. We implement four different methods: a mathematical formulation that can be solved using an IP solver, a branch-and-bound algorithm, a beam search procedure and a tabu search method. Lagrangian relaxation is embedded in the algorithms for computing lower bounds. The different solution frameworks are compared via extensive computational experiments.

Comparison of Integer Programming (IP) Solvers for Automated Test Assembly (ATA)

At the heart of van der Linden's approach to automated test assembly (ATA) is a linear programming/integer programming (LP/IP) problem. A variety of IP solvers are available, ranging in cost from free to hundreds of thousands of dollars. In this paper, I compare several approaches to solving the underlying IP problem. These approaches range from traditional computer programming, through LP/IP‐specific modeling languages, to plug‐ins for common software such as Excel. The features of several of the major IP solvers are briefly reviewed, describing which of the features are more or less useful in the context of ATA. The appendices include a list of resources that I have found particularly useful.

Virtual private network design over the first Chvátal closure

In this paper we consider the virtual private network (VPN) design problem. Given upper bounds on the amount of traffic that an endpoint could send or receive, the problem needs to reserve enough capacities in such a way that any demand matrix that respects the upper bound could be routed without exceeding the reserved capacities and the total reservation cost is minimized. In On the difficulty of virtual private network instances (Networks 63 (2014) 327–333), we argued that the computational investigation on exact mathematical programming approaches for VPN needs to be revised after that challenging instances have been exposed. To that end, we consider the VPN design problem over the first Chvatal closure and demonstrate that tight solutions could be found for the VPN design problem only by optimizing over the closure. First, we perform theoretical investigation on adding rank-1 Chvatal–Gomory cuts to the problem. Along the way, an important property for such cuts is proved that omits a large number of redundant rank-1 cuts. We then provide interesting insights about the problem and reduce the existing MIP formulations to a binary one. On the computational side, we investigate the idea of adding rank-1 cuts more aggressively in order to computationally evaluate tightness of the first Chvatal closure for the VPN design problem. Here, the binary reduction plays an important role allowing the use of special cuts of the first closure, namely the zero-half cuts. We show that, almost all the success of the first Chvatal closure of the VPN design problem in raising dual bound is due to zero-half cuts. Our experiments on the benchmark instances in this article show that a state-of-the-art IP solver without using zero-half cuts could not even hit the challenging benchmarks. As a results a cut-and-branch framework that aggressively adds such cuts at the root could solve the challenging VPN instances to the extent of zero or small integrality gap in a reasonable amount of time.

A new cut-and-solve and cutting plane combined approach for the capacitated lane reservation problem

This paper investigates a capacitated lane reservation problem with residual capacity. The focus of the problem is to design time-guaranteed paths for a set of transportation tasks via lane reservation with respect to the residual capacity issue. The lane reservation can ensure the time-guaranteed travel but has negative impact on the normal traffic. Thus, the objective of the problem is to minimize the impact of such lane reservation by optimally selecting lanes in the network to be reserved. For this NP-hard problem, an exact approach based on the cut-and-solve and cutting plane methods is developed. New separation algorithms are proposed to find appropriate valid inequalities according to the characteristic of the problem to accelerate the optimal convergence of the resolution approach. Numerical computational results show that the developed approach is efficient as compared with state-of-the-art algorithms in the literature and the IP solver CPLEX.

Puzzle—IP Satisfied

Finding a feasible as opposed to optimal solution of an integer program model comes under the heading of constraint programming and if a model is also a binary integer program (BIP) it is known to be equivalent to the satisfiability (SAT) problem from Boolean logic. Many of the puzzles treated in this column over the years and a host of others fall into this category, that is, they may be modeled purely using binary variables and the aim is to fulfil the conditions of the puzzle rather than to optimize a function. This leads to the possibility that there may be gains in efficiency to be found by looking beyond general IP solvers to more specialized programs. The following article presents five puzzles together with IP formulations where the writer has experienced significant improvements in solution times using a SAT solver rather than a general purpose IP solver. The puzzles will be modeled and solved using three freely available open source packages, namely GLPK, Minisat and SCIP. GLPK and Minisat are both available in the GUSEK distribution from http://gusek.sourceforge.net/gusek.html and SCIP is available from http://scip.zib.de/ . GLPK is a general purpose linear and mixed integer solver, SCIP solves both Integer Programs and Constraint Programs and Minisat is exclusively an SAT solver. The GUSEK environment provides direct access to GLPK and Minisat. It also has the option to generate MPS formatted files of Mathprog models which may be read into SCIP and solved.

Exact algorithms for a generalization of the order acceptance and scheduling problem in a single-machine environment

This paper studies a generalization of the order acceptance and scheduling problem in a single-machine environment where a pool consisting of firm planned orders as well as potential orders is available from which an over-demanded company can select. The capacity available for processing the accepted orders is limited and each order is characterized by a known processing time, delivery date, revenue and a weight representing a penalty per unit-time delay beyond the delivery date. We prove that the existence of a constant-factor approximation algorithm for this problem is unlikely. We propose two linear formulations that are solved using an IP solver and we devise two exact branch-and-bound procedures able to solve instances with up to 50 jobs within reasonable CPU times. We compare the efficiency and quality of the results obtained using the different solution approaches.

MABS: Spreadsheet-based decision support for precision marketing

In this paper, we describe a decision support system developed for automatically scheduling and optimising broadcasts of advertisements to mobile phones via SMS (Short Message Service) text messaging. The system, MABS or “Mobile Advertising Broadcast Scheduler”, is developed in Microsoft Excel with a link to Lingo, a modelling language and IP solver. It was developed for a London-based company specialized in location-sensitive precision marketing via mobile phones. The system significantly reduced the time required to schedule the broadcasts, and resulted both in increased customer response and revenues.

An integer programming formulation for a case study in university timetabling

A novel 0–1 integer programming formulation of the university timetabling problem is presented. The model provides constraints for a great number of operational rules and requirements found in most academic institutions. Treated as an optimization problem, the objective is to minimize a linear cost function. With this objective, it is possible to consider the satisfaction of expressed preferences regarding teaching periods or days of the week or even classrooms for specified courses. Moreover, with suitable definition of the cost coefficients in the objective function it is possible to reduce the solution space and make the problem tractable. The model is solvable by existing software tools with IP solvers, even for large departments. The case of a five-year Engineering Department with a large number of courses and teachers is presented along with its solution as resulted from the presented IP formulation.

Solving a Practical Pickup and Delivery Problem

We consider a pickup and delivery vehicle routing problem commonly encountered in real-world logistics operations. The problem involves a set of practical complications that have received little attention in the vehicle routing literature. In this problem, there are multiple carriers and multiple vehicle types available to cover a set of pickup and delivery orders, each of which has multiple pickup time windows and multiple delivery time windows. Orders and carrier/vehicle types must satisfy a set of compatibility constraints that specify which orders cannot be covered by which carrier/vehicle types and which orders cannot be shipped together. Order loading and unloading sequence must satisfy the nested precedence constraint that requires that an order cannot be unloaded until all the orders loaded into the truck later than this order are unloaded. Each vehicle trip must satisfy the driver's work rules prescribed by the Department of Transportation which specify legal working hours of a driver. The cost of a trip is determined by several factors including a fixed charge, total mileage, total waiting time, and total layover time of the driver. We propose column generation based solution approaches to this complex problem. The problem is formulated as a set partitioning type formulation containing an exponential number of columns. We apply the standard column generation procedure to solve the linear relaxation of this set partitioning type formulation in which the resulting master problem is a linear program and solved very efficiently by an LP solver, while the resulting subproblems are computationally intractable and solved by fast heuristics. An integer solution is obtained by using an IP solver to solve a restricted version of the original set partitioning type formulation that only contains the columns generated in solving the linear relaxation. The approaches are evaluated based on lower bounds obtained by solving the linear relaxation to optimality by using an exact dynamic programming algorithm to solve the subproblems exactly. It is shown that the approaches are capable of generating near-optimal solutions quickly for randomly generated instances with up to 200 orders. For larger randomly generated instances with up to 500 orders, it is shown that computational times required by these approaches are acceptable.

Solving a Truck Dispatching Scheduling Problem Using Branch-and-Cut

A branch-and-cut IP solver is developed for a class of structured 0/1 integer programs arising from a truck dispatching scheduling problem. This problem involves a special class of knapsack equality constraints. Families of facets for the polytopes associated with individual knapsack constraints are identified. In addition, a notion of “conflict graph” is utilized to obtain an approximating node-packing polytope for the convex hull of all 0/1 solutions. The branch-and-cut solver generates cuts based on both the knapsack equality constraints and the approximating node-packing polytope, and incorporates these cuts into a tree-search algorithm that uses problem reformulation and linear programming-based heuristics at each node in the search tree to assist in the solution process. Numerical experiments are performed on large-scale real instances supplied by Texaco Trading & Transportation, Inc. The optimal schedules correspond to cost savings for the company and greater job satisfaction for drivers due to more balanced work schedules and income distribution.