Allocation of Classroom Space Using Linear Programming (A Case Study: Premier Nurses Training College, Kumasi)

The use of linear programming to solve the problem of over-allocation and under-allocation of the scarce classroom space was considered with particular reference to the Premier Nurse’s Training College, Kumasi. Data was collected from the College on the classroom facilities and the number of students per programme. A linear programming model was formulated based on the data collected to maximize the usage of the limited classroom space. POM-QM for Windows 4 (Software for Quantitative Methods, Production and Operation Management by Howard J. Weiss) was used based on the simplex algorithm to obtain optimal solution.Analysis of the results showed that six (50%) of the twelve classrooms could be used to create a maximum classroom space of six hundred and forty.  It was also observed that the management could use two hundred and eighty (280) surplus spaces to increase its student’s intake from three hundred and sixty (360) to six hundred and forty (640) students, an increase of about 77.78% with only 50% of the total number of classrooms. Again management could cut down the number of classrooms used from twelve to six and reduce the cost of maintaining the classrooms by 50% and still have as many as six extra classrooms for other equally important purposes, hence maximize its profit margin. Keywords: Linear programming, Allocation, Optimal solution, Simplex Algorithm, Premier Nurse’s Training College.

Cut generation for an integrated employee timetabling and production scheduling problem

International audienceThis paper investigates the integration of the employee timetabling and production scheduling problems. At the first level, we manage a classical employee timetabling problem. At the second level, we aim at supplying a feasible production schedule for a set of interruptible tasks with qualification requirements and time-windows. Instead of hierarchically solving these two problems as in the current practice, we try here to integrate them and propose two exact methods to solve the resulting problem. The former is based on a Benders decomposition while the latter relies on a specific decomposition and a cut generation process. The relevance of these different approaches is discussed here through experimental results

A Flexible Model and a Hybrid Exact Method for Integrated Employee Timetabling and Production Scheduling

We propose a flexible model and several integer linear programming and constraint programming formulations for integrated employee timetabling and production scheduling problems. A hybrid constraint and linear programming exact method is designed to solve a basic integrated employee timetabling and job-shop scheduling problem for lexicographic minimization of makespan and labor costs. Preliminary computational experiments show the potential of hybrid methods

Integer linear programs and heuristic solution approaches for different planning levels in underground mining

Natürlich vorkommende Mineralien werden seit Tausenden von Jahren aus der Erde gefördert. Im Bergbau wird Operations Research (OR) hauptsächlich angewendet, um die Materialgewinnung zu vereinfachen und die Ressourcen für die Gewinnung effizienter zuzuordnen. Optimierungsprobleme im Bergbau werden üblicherweise nach ihrem Planungshorizont eingeordnet. Dabei werden Layout- und Designprobleme auf strategischer, Produktions- und Planungsprobleme auf taktischer und Ressourcenzuordnungsprobleme auf operativer Planungsebene behandelt. In dieser kumulativen Dissertation betrachten wir eine der größten deutschen Kalibergwerke und befassen uns mit drei Optimierungsproblemen auf drei verschiedenen Planungsebenen. Zunächst betrachten wir eine sogenannte „Gewinnungsprogrammplanung“ für einen Planungshorizont von einem Monat auf taktischer Planungsebene. Die betrachtete qualitätsorientierte Zielfunktion zielt auf eine gleichmäßige Kalisalzgewinnung hinsichtlich des beinhalteten Kaliums ab. Da die Menge der Gesamtförderung a priori unbekannt ist, kann die in der Gesamtförderung enthaltene Kaliummenge mithilfe nicht-linearer Nebenbedingungen in der mathematischen Formulierung bestimmt werden. Die Herausforderung besteht in der Linearisierung der entsprechenden Nebenbedingungen, damit ein gemischt ganzzahliges lineares Programm eingeführt werden kann. Darüber hinaus schlagen wir eine Heuristik vor, welche mindestens eine zulässige Lösung für realitätsnahe Probleminstanzen innerhalb eines angemessenen Zeitraums findet. Die Performanceanalyse an 100 zufällig generierten Probleminstanzen zeigt, dass eine subtile Kombination des vorgeschlagenen mathematischen Programms mit der eingeführten Heuristik nahezu optimale Lösungen für praxisrelevante Probleme findet. Als Nächstes betrachten wir eine „Grobplanung des Maschineneinsatzes“ innerhalb eines Planungshorizonts von einer Woche, welche zwischen der taktischen und der operativen Planungsebene eingeordnet werden kann und untersucht, ob die Ergebnisse der Gewinnungsprogrammplanung für die erste Woche des folgenden Monats umgesetzt werden können. Hierzu wird ein Maschinenplanungsproblem zur Minimierung des maximalen Fertigstellungszeitpunkts berücksichtigt. Wir stellen ein gemischt ganzzahliges lineares Programm vor, welches bestimmte Umstände in einem untertägigen Bergwerk wie die Wiederholung der Erstfreigabe berücksichtigt. Die größte Herausforderung besteht darin, einen Lösungsansatz zu entwickeln, der nahezu optimale Lösungen für große Probleminstanzen findet. Also wird eine Heuristik vorgeschlagen, der absichtliche Verzögerungen von Jobs vor Bearbeitungsstufen einbezieht, d. h. sogenannte aktive Pläne erzeugt. Die Performanceanalyse zeigt, dass kleine Probleminstanzen mit CPLEX optimal gelöst werden können. Bei größeren Instanzen liefert die vorgeschlagene Heuristik die besten Ergebnisse. Schließlich wird auf der operativen Planungsebene eine „Feinplanung des Maschinen- und Personaleinsatzes“ berücksichtigt. Das betrachtete Problem verfolgt einen gleichmäßigen Fortschritt im untertägigen Bergwerk innerhalb einer Arbeitsschicht. Um realistische Lösungen zu erstellen, müssen verschiedene Arten von Rüstzeiten in Betracht gezogen werden, die abhängig von der Bearbeitungsreihenfolge der Operationen an Maschinen und Arbeitern entstehen. Die größte Herausforderung besteht darin, die spezifischen Umstände einer Arbeitsschicht mathematisch darzustellen, z. B. die Berücksichtigung der Pausen der Mitarbeiter für eine eventuelle Verzögerung der Bearbeitungszeit, das Bestimmen des bearbeiteten Prozentsatzes eines Jobs während einer Arbeitsschicht, die Berechnung der Entfernungs- und Umrüstzeiten usw. Wir stellen eine Heuristik vor, die aus zwei Schritten besteht. Im ersten Schritt wird eine Relaxation des Problems unter Einhaltung einen Teil der genannten Nebenbedingungen gelöst. Die gefundene, typischerweise unzulässige Lösung wird im zweiten Schritt durch Einfügen der vernachlässigten Zeiten repariert. Die Ergebnisse zeigen, dass die vorgeschlagene Heuristik für 70 Prozent der realitätsnahen Probleminstanzen eine bessere Lösung als eine bestehende Heuristik finden kann. Anschließend formulieren wir ein neues, kompaktes, gemischt ganzzahliges lineares Programm, das mithilfe von TSP-Variablen alle Problemspezifikationen berücksichtigt. Wir zeigen, dass das vorgeschlagene gemischt ganzzahlige lineare Programm die vorgeschlagene zweistufige Heuristik erheblich übertrifft.Humans have been extracting naturally occurring minerals from the earth for thousands of years. In mining, operations research (OR) has been mainly used to help the mine planners decide how the material can be extracted and what to do with the material removed, what kind of resources to use for the extraction, and how to allocate the resources. It is very widespread to classify decision problems according to their time horizons, where 1. layout and design problems, 2. production and scheduling problems, and 3. operational equipment allocation problems are considered on strategic, tactical, and operational planning levels, respectively. In this cumulative dissertation thesis, we consider one of the biggest German potash mines and address three optimization problems on three different planning levels. First, we consider a so-called “extraction program planning” for a time horizon of one month on the tactical planning level. The related quality-oriented objective function aims at an even extraction of potash regarding the potassium content. For mathematically formulating the objective function, the amount of potassium contained in the output must be determined. Since the amount of total output is a priori unknown, the potassium amount can be determined primarily using non-linear constraints. The principal challenge is the linearization of the corresponding constraints to introduce a mixedinteger linear program with a quality-related objective function. We also propose a heuristic solution procedure that finds for realistically-sized problem instances at least one feasible solution within a reasonable amount of time. The performance analysis conducted on 100 randomly generated problem instances shows that a sophisticated combination of the proposed mixed-integer linear program and the introduced heuristic approach finds high-quality, near-optimal solutions for practice-relevant problems. Next, we deal with a “preliminary (conceptual) planning of machines” within a time horizon of one week. That problem can be classified between the tactical and operational planning levels and investigates whether the results of the extraction program planning can be implemented for the first week of the following month. For this purpose, a machine scheduling problem to minimize the makespan is taking into account. We propose a mixed-integer linear program considering particular circumstances in an underground mine, e.g., reentry. The main challenge is to provide a solution approach that can find near-optimal solutions for large-sized problem instances. For this purpose, we suggest a heuristic approach considering conscious delays of jobs in front of production stages, i.e., active scheduling is applied. The performance analysis shows that small problem instances can be optimally solved with CPLEX-solver. For larger problem instances, the best performance is achieved by the suggested advanced multi-start heuristic. Finally, a “detailed shift planning” considering a simultaneous assignment of machines and workers is taken into account on the operational planning level. That problem pursues an even progress in the underground mine within a work shift. During a work shift, in addition to a machine scheduling problem, a personnel allocation problem must be considered. Moreover, to provide realistic solutions, different kinds of setup times must be observed, depending on the processing sequence of the operations on machines and workers. The major challenge is to express the specific circumstances of a work shift mathematically, e.g., considering workers' breaks for a possible delay in the processing time of a job, determining the processed percentage of a job during a work shift, observing removal and changeover times, etc. A part of real constraints is formulated in a relaxed program as part of a heuristic solution approach. The proposed heuristic procedure consists of two steps. In the first step, a relaxed program neglecting some setup times is solved, and the typically unfeasible solution achieved is repaired in the second step by inserting the neglected times. The results show that the proposed heuristic can find for 70 percent of the realistic problem instances a better solution than an existing heuristic approach. Subsequently, we introduce a new, compact mixed-integer linear program using TSPvariables considering all the problem specifications. We show that the proposed mixed-integer linear program outperforms the proposed two-stage heuristic considerably

No-wait open shop com flexibilidade de operadores

Orientador: Prof. Dr. Cassius Tadeu ScarpinTese (doutorado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa : Curitiba, 29/09/2021Inclui referências: p. 97-103Área de concentração: Programação MatemáticaResumo: O Problema de Open Shop pode ser definido como um problema de Programação da Produção, em que os itens não possuem rotas de processamento pré-definidas. A implicação da não existência de sequencias pré-definidas gera, para o modelo matemático de Programação Linear Inteira Mista (PLIM) proposto, o aumento do espaço de solução e, por conseguinte, a complexidade em encontrar soluções para o problema. Aplicações de Open Shop são encontradas, por exemplo, em centros de exames médicos e laboratoriais, oficinas mecânicas, centros de controle de qualidade, entre outros. Quando a produção de itens se dá de modo contínuo, tem-se a produção denominada sem espera (no-wait), verificada em ambientes cujas propriedades físicoquímicas do material processado são alteradas caso haja espera entre as máquinas, como na indústria farmacêutica. Esse trabalho visa apresentar um modelo PLIM que aborde a temática do Open Shop com restrições de flexibilidade de operadores e de no-wait, a fim de minimizar o tempo de fluxo total, expresso pela diferença entre o instante de término do processamento do item e a data de liberação deste. Propõe-se uma ponderação em relação à restrição de recursos, no caso a mão de obra, flexível e multifacetada, e a restrição de no-wait, relacionada a produção contínua de itens. Com o propósito de validar o modelo, diversos cenários foram criados, testando-se diferentes níveis de flexibilidade dos operadores, avaliando o impacto desta na obtenção de maior factibilidade do problema. Os resultados obtidos refletem que o aumento de apenas uma habilidade pode reduzir sensivelmente a inviabilidade do modelo. Cenários em que o operador domina apenas uma habilidade possuem infactibilidade próxima a 50%. Além disso, devido à complexidade resolutiva do modelo proposto, instâncias com mais de cinco itens majoritariamente nem sequer apresentam solução inicial válida dentro do tempo limite proposto.Abstract: The Open Shop Problem can be defined as a Production Scheduling problem, in which the items do not have predefined processing routes. The implication of the nonexistence of predefined sequences generates, for the proposed MILP (Mixed Integer Linear Programming) Problem, the increase of the solution space and, therefore, the complexity of finding solutions for the problem. Applications of Open Shop are found, for example, in medical and laboratory examination centers, mechanical workshops, quality control centers, among others. When the job production is continuous, we have no-wait processing, which is encountered in environments where the physical and chemical properties of the processed material are altered if there is wait between machines, such as pharmaceutical industry. This work aims to present a MILP model which addresses the Open Shop theme with operator flexibility and no-wait in order to minimize the total flowtime, expressed by the difference between completion time of a job and its release date. We propose weighing with regards to the resource constraints, in the case of flexible and multi-skilled labor, and no-wait, related to job continuous production. With respect to validate the model, several scenarios were created, addressing different degrees of worker flexibility, assessing its impact on problem feasibility. The obtained results highlight that the addition of only one skill can pronouncedly reduce the model unavailability. Scenarios where the worker masters only one skill possess infeasibility levels close to 50%. Furthermore, due to the solving complexity of the proposed model, benchmarks with more than five jobs mostly not even present a valid initial solution within the designed time limit