Search results for: mixed integer programming

Authors: A. Bari

Abstract:

Integer programming is a branch of mathematical programming techniques in operations research in which some or all of the variables are required to be integer valued. Various cuts have been used to solve these problems. We have also developed cuts known as NAZ cut & A-T cut to solve the integer programming problems. These cuts are used to reduce the feasible region and then reaching the optimal solution in minimum number of steps.

Keywords: Integer Programming, NAZ cut, A-T cut, Cutting plane method

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Authors: Vijaya K. Srivastava, Davide Spinello

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This paper presents established 3ⁿ enumeration procedure for mixed integer optimization problems for solving multi-objective reliability and redundancy allocation problem subject to design constraints. The formulated problem is to find the optimum level of unit reliability and the number of units for each subsystem. A number of illustrative examples are provided and compared to indicate the application of the superiority of the proposed method.

Keywords: integer programming, mixed integer programming, multi-objective optimization, Reliability Redundancy Allocation

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Authors: Mohsen Ziaee

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In this paper, a mixed integer linear programming (MILP) model is presented to solve the flexible job shop scheduling problem (FJSP). This problem is one of the hardest combinatorial problems. The objective considered is the minimization of the makespan. The computational results of the proposed MILP model were compared with those of the best known mathematical model in the literature in terms of the computational time. The results show that our model has better performance with respect to all the considered performance measures including relative percentage deviation (RPD) value, number of constraints, and total number of variables. By this improved mathematical model, larger FJS problems can be optimally solved in reasonable time, and therefore, the model would be a better tool for the performance evaluation of the approximation algorithms developed for the problem.

Keywords: scheduling, flexible job shop, makespan, mixed integer linear programming

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Authors: Chaimongkol Limpianchob

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This work addresses the problem of production planning that arises in the production of aromatic coconuts from Samudsakhorn province in Thailand. The planning involves the forwarding of aromatic coconuts from the harvest areas to the factory, which is classified into two groups; self-owned areas and contracted areas, the decisions of aromatic coconuts flow in the plant, and addressing a question of which warehouse will be in use. The problem is formulated as a mixed-integer linear programming model within supply chain management framework. The objective function seeks to minimize the total cost including the harvesting, labor and inventory costs. Constraints on the system include the production activities in the company and demand requirements. Numerical results are presented to demonstrate the feasibility of coconuts supply chain model compared with base case.

Keywords: aromatic coconut, supply chain management, production planning, mixed-integer linear programming

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Authors: Harshith Gowda K. S, Tejaskumar N, Shubhanga R. B, Gowtham N, Deekshith Gowda H. S

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Phasor measurement units (PMU) are playing an important role in the current power system for state estimation. It is necessary to have complete observability of the power system while minimizing the cost. For this purpose, the optimal location of the phasor measurement units in the power system is essential. In a bus system, zero injection buses need to be evaluated to minimize the number of PMUs. In this paper, the optimization problem is formulated using mixed integer programming to obtain the optimal location of the PMUs with increased observability. The formulation consists of with and without zero injection bus as constraints. The formulated problem is simulated using a CPLEX solver in the GAMS software package. The proposed method is tested on IEEE 30, IEEE 39, IEEE 57, and IEEE 118 bus systems. The results obtained show that the number of PMUs required is minimal with increased observability.

Keywords: PMU, observability, mixed integer programming (MIP), zero injection buses (ZIB)

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Authors: K. -H. Yang

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Supplier selection problem is one of the important issues of supply chain problems. Two categories of methodologies include qualitative and quantitative approaches which can be applied to supplier selection problems. However, due to the complexities of the problem and lacking of reliable and quantitative data, qualitative approaches are more than quantitative approaches. This study considers operational cost and supplier’s reliability factor and solves the problem by using a quantitative approach. A mixed integer programming model is the primary analytic tool. Analyses of different scenarios with variable cost and reliability structures show that the effectiveness of this approach to the supplier selection problem.

Keywords: mixed integer programming, quantitative approach, supplier’s reliability, supplier selection

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Authors: Cem Isik Dogru, Salih Tekin, Kursad Derinkuyu

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Electric vehicle (EV) carsharing is an alternative method to tackle urban transportation problems. This method can be applied by several options. One of the options is the one-way carsharing, which allow an EV to be taken at a designated location and leaving it on another specified location customer desires. Although it may increase users’ satisfaction, the issues, namely, demand dissatisfaction, relocation of EVs and charging schedules, must be dealt with. Also, excessive electricity has to be stored in batteries of EVs. To cope with aforementioned issues, two-step mixed integer programming (MIP) model is proposed. In first step, the integer programming model is used to determine amount of electricity to be sold to the grid in terms of time periods for extra profit. Determined amounts are provided from the batteries of EVs. Also, this step works in day-ahead electricity markets with forecast of periodical electricity prices. In second step, other MIP model optimizes daily operations of one-way carsharing: charging-discharging schedules, relocation of EVs to serve more demand and renting to maximize the profit of EV fleet owner. Due to complexity of the models, heuristic methods are introduced to attain a feasible solution and different price information scenarios are compared.

Keywords: electric vehicles, forecasting, mixed integer programming, one-way carsharing

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Authors: Majid Khalili, Hamed Tayebi

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This paper considers a stochastic flexible job-shop scheduling (SFJSS) problem in the presence of production disruptions, and reactive scheduling is implemented in order to find the optimal solution under uncertainty. In this problem, there are two main disruptions including machine failure which influences operation time, and modification or cancellation of the order delivery date during production. In order to decrease the negative effects of these difficulties, two derived strategies from reactive scheduling are used; the first one is relevant to being able to allocate multiple machine to each job, and the other one is related to being able to select the best alternative process from other job while some disruptions would be created in the processes of a job. For this purpose, a Mixed Integer Linear Programming model is proposed.

Keywords: flexible job-shop scheduling, reactive scheduling, stochastic environment, mixed integer linear programming

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Authors: Quoc Khanh Dang, Thomas Bourdeaud’huy, Khaled Mesghouni, Armand Toguy´eni

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This paper studies a train routing and scheduling problem for busy railway stations. Our objective is to allow trains to be routed in dense areas that are reaching saturation. Unlike traditional methods that allocate all resources to setup a route for a train and until the route is freed, our work focuses on the use of resources as trains progress through the railway node. This technique allows a larger number of trains to be routed simultaneously in a railway node and thus reduces their current saturation. To deal with this problem, this study proposes an abstract model and a mixed-integer linear programming formulation to solve it. The applicability of our method is illustrated on a didactic example.

Keywords: busy railway stations, mixed-integer linear programming, offline railway station management, train platforming, train routing, train scheduling

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Authors: Ameen Alawneh

Abstract:

The cost of material management across the supply chain represents a major contributor to the overall cost of goods in many companies both manufacturing and service sectors. This fact combined with the fierce competition make supply chains more efficient and cost effective. It also requires the companies to improve the quality of the products and services, increase the effectiveness of supply chain operations, focus on customer needs, reduce wastes and costs across the supply chain. As a heavy industry, steel manufacturing companies in particular are nowadays required to be more environmentally conscious due to their contribution to air, soil, and water pollution that results from emissions and wastes across their supply chains. Steel companies are increasingly looking for methods to reduce or cost cut in the operations and provide extra value to their customers to stay competitive under the current low margins. In this research we develop a green framework model for the sustainability of a steel company supply chain using Mixed integer Linear programming.

Keywords: Supply chain, Mixed Integer linear programming, heavy industry, water pollution

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Authors: Angelina Anani, Ignacio Ortiz Flores, Haitao Li

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The use of underground mining methods have increased significantly over the past decades. This increase has also been spared on by several mines transitioning from surface to underground mining. However, determining the transition depth can be a challenging task, especially when coupled with production schedule optimization. Several researchers have simplified the problem by excluding operational features relevant to production schedule optimization. Our research objective is to investigate the extent to which operational features of transition mines accounted for affect the optimal production schedule. We also provide a framework for factors to consider in production schedule optimization for transition mines. An integrated mixed-integer linear programming (MILP) model is developed that maximizes the NPV as a function of production schedule and transition depth. A case study is performed to validate the model, with a comparative sensitivity analysis to obtain operational insights.

Keywords: underground mining, transition mines, mixed-integer linear programming, production schedule

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Authors: Didem Can

Abstract:

Scheduling problems are one of the most fundamental issues of production systems. Many different approaches and models have been developed according to the production processes of the parts and the main purpose of the problem. In this study, one of the bottleneck stations of a company serving in the aerospace industry is analyzed and considered as a single machine scheduling problem with sequence-dependent setup times. The objective of the problem is assigning a large number of similar parts to the same shift -to reduce chemical waste- while minimizing the number of tardy jobs. The goal programming method will be used to achieve two different objectives simultaneously. The assignment of parts to the shift will be expressed using the integer programming method. Finally, the constraint programming method will be used as it provides a way to find a result in a short time by avoiding worse resulting feasible solutions with the defined variables set. The model to be established will be tested and evaluated with real data in the application part.

Keywords: constraint programming, goal programming, integer programming, sequence-dependent setup, single machine scheduling

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Authors: Nur Aidya Hanum Aizam, Vikneswary Uvaraja

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The agenda of showing the scheduled time for performing certain tasks is known as timetabling. It widely used in many departments such as transportation, education, and production. Some difficulties arise to ensure all tasks happen in the time and place allocated. Therefore, many researchers invented various programming model to solve the scheduling problems from several fields. However, the studies in developing the general integer programming model for many timetabling problems are still questionable. Meanwhile, this thesis describe about creating a general model which solve different types of timetabling problems by considering the basic constraints. Initially, the common basic constraints from five different fields are selected and analyzed. A general basic integer programming model was created and then verified by using the medium set of data obtained randomly which is much similar to realistic data. The mathematical software, AIMMS with CPLEX as a solver has been used to solve the model. The model obtained is significant in solving many timetabling problems easily since it is modifiable to all types of scheduling problems which have same basic constraints.

Keywords: AIMMS mathematical software, integer linear programming, scheduling problems, timetabling

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Authors: M. Babul Hasan

Abstract:

Management sciences (MS), big group of companies and industries or government policies (GP) is affiliated with a huge number of decision ingredients and complicated restrictions. Every factor in MS, every product in Industries or decision in GP is not always bankable in practice. After formulating these models there arises large-scale mixed integer programming (MIP) problem. In this paper, we developed decomposition-based pricing procedure to filter the unnecessary decision ingredients from MIP where the variables in huge number will be abated and the complicacy of restrictions will be elementary. A real life numerical example has been illustrated to demonstrate the methods. We develop the computer techniques for these methods by using a mathematical programming language (AMPL).

Keywords: Lagrangian relaxation, decomposition, sub-problem, master-problem, pricing, mixed IP, AMPL

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Authors: Babak H. Tabrizi, Seyed Farid Ghaderi

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Concurrent planning of project scheduling and material ordering can provide more flexibility to the project scheduling problem, as the project execution costs can be enhanced. Hence, the issue has been taken into account in this paper. To do so, a mixed-integer mathematical model is developed which considers the aforementioned flexibility, in addition to the materials quantity discount and space availability restrictions. Moreover, the activities duration has been treated as decision variables. Finally, the efficiency of the proposed model is tested by different instances. Additionally, the influence of the aforementioned parameters is investigated on the model performance.

Keywords: material ordering, project scheduling, quantity discount, space availability

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Authors: Oscar Javier Herrera Ochoa, Ivan Dario Romero Fonseca

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This paper presents a design of a model for planning the distribution logistics operation. The significance of this work relies on the applicability of this fact to the analysis of small and medium enterprises (SMEs) of dry freight in Bogotá. Two stages constitute this implementation: the first one is the place where optimal planning is achieved through a hybrid model developed with mixed integer programming, which considers the transhipment operation based on a combined load allocation model as a classic transshipment model; the second one is the specific routing of that operation through the heuristics of Clark and Wright. As a result, an integral model is obtained to carry out the step by step planning of the distribution of dry freight for SMEs in Bogotá. In this manner, optimum assignments are established by utilizing transshipment centers with that purpose of determining the specific routing based on the shortest distance traveled.

Keywords: transshipment model, mixed integer programming, saving algorithm, dry freight transportation

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Authors: Ahmed Al-Sultan

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The check-in area of airport terminal is one of the busiest sections at airports at certain periods. The passengers are subjected to queues and delays during the check-in process. These delays and queues are due to constraints in the capacity of service facilities. In this project, the airport terminal is decomposed into several check-in areas. The airport check-in scheduling problem requires both a deterministic (integer programming) and stochastic (simulation) approach. Integer programming formulations are provided to minimize the total number of counters in each check-in area under the realistic constraint that counters for one and the same flight should be adjacent and the desired number of counters remaining in each area should be fixed during check-in operations. By using simulation, the airport system can be modeled to study the effects of various parameters such as number of passengers on a flight and check-in counter opening and closing time.

Keywords: airport terminal, integer programming, scheduling, simulation

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Authors: Ebrahim Asadi-Gangraj

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Hybrid flow shop scheduling problem (HFS) contains sequencing in a flow shop where, at any stage, there exist one or more related or unrelated parallel machines. This production system is a common manufacturing environment in many real industries, such as the steel manufacturing, ceramic tile manufacturing, and car assembly industries. In this research, a mixed integer linear programming (MILP) model is presented for the hybrid flow shop scheduling problem, in which, the objective consists of minimizing the maximum completion time (makespan). For this purpose, a Benders Decomposition (BD) method is developed to solve the research problem. The proposed approach is tested on some test problems, small to moderate scale. The experimental results show that the Benders decomposition approach can solve the hybrid flow shop scheduling problem in a reasonable time, especially for small and moderate-size test problems.

Keywords: hybrid flow shop, mixed integer linear programming, Benders decomposition, makespan

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Authors: Mohammadali Abedini Sanigy, Jiangang Fei

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In this paper, a mathematical optimization model was developed to maximize the profit in a vegetable production planning problem. It serves as a decision support system that assists farmers in land allocation to crops and harvest scheduling decisions. The developed model can handle different rotation rules in two consecutive cycles of production, which is a common practice in organic production system. Moreover, different production methods of the same crop were considered in the model formulation. The main strength of the model is that it is not restricted to predetermined production periods, which makes the planning more flexible. The model is classified as a mixed integer programming (MIP) model and formulated in PYOMO -a Python package to formulate optimization models- and solved via Gurobi and CPLEX optimizer packages. The model was tested with secondary data from 'Australian vegetable growing farms', and the results were obtained and discussed with the computational test runs. The results show that the model can successfully provide reliable solutions for real size problems.

Keywords: crop rotation, harvesting, mathematical model formulation, vegetable production

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Authors: Ebert Brea

Abstract:

We propose a novel direct search algorithm for identifying at least a local minimum of mixed integer nonlinear unconstrained optimization problems. The Mixed Integer Randomized Pattern Search Algorithm (MIRPSA), so-called by the author, is based on a randomized pattern search, which is modified by the MIRPSA for finding at least a local minimum of our problem. The MIRPSA has two main operations over the randomized pattern search: moving operation and shrinking operation. Each operation is carried out by the algorithm when a set of conditions is held. The convergence properties of the MIRPSA is analyzed using a Markov chain approach, which is represented by an infinite countable set of state space λ, where each state d(q) is defined by a measure of the qth randomized pattern search Hq, for all q in N. According to the algorithm, when a moving operation is carried out on the qth randomized pattern search Hq, the MIRPSA holds its state. Meanwhile, if the MIRPSA carries out a shrinking operation over the qth randomized pattern search Hq, the algorithm will visit the next state, this is, a shrinking operation at the qth state causes a changing of the qth state into (q+1)th state. It is worthwhile pointing out that the MIRPSA never goes back to any visited states because the MIRPSA only visits any qth by shrinking operations. In this article, we describe the MIRPSA for mixed integer nonlinear unconstrained optimization problems for doing a deep study of its convergence properties using Markov chain viewpoint. We herein include a low dimension case for showing more details of the MIRPSA, when the algorithm is used for identifying the minimum of a mixed integer quadratic function. Besides, numerical examples are also shown in order to measure the performance of the MIRPSA.

Keywords: direct search, mixed integer optimization, random search, convergence, Markov chain

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Authors: Taehan Lee

Abstract:

We consider a network design problem which has shortest routing restriction based on the values determined by the installed facilities on each arc. In conventional multicommodity network design problem, a commodity can be routed through any possible path when the capacity is available. But, we consider a problem in which the commodity between two nodes must be routed on a path which has shortest metric value and the link metric value is determined by the installed facilities on the link. By this routing restriction, the problem has a distinct characteristic. We present an integer programming formulation containing the primal-dual optimality conditions to the shortest path routing. We give some computational results for the model.

Keywords: integer programming, multicommodity network design, routing, shortest path

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Authors: Ruben Lopez-Rodriguez

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In a microgrid, diverse energy sources (both renewable and non-renewable) are combined with energy storage units to form a localized power system. Microgrids function as independent entities, capable of meeting the energy needs of specific areas or communities. This paper introduces a Model Predictive Control (MPC) approach tailored for grid-connected microgrids, aiming to optimize their operation. The formulation employs Mixed-Integer Programming (MIP) to find optimal trajectories. This entails the fulfillment of continuous and binary constraints, all while accounting for commutations between various operating conditions such as storage unit charge/discharge, import/export from/towards the main grid, as well as asset connection/disconnection. To validate the proposed approach, a microgrid case study is conducted, and the simulation results are compared with those obtained using a rule-based strategy.

Keywords: microgrids, mixed logical dynamical systems, mixed-integer optimization, model predictive control

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Authors: Y. Long, L. Liu, K. V. Branin

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One challenge faced by procurement decision-maker during the acquisition process is how to compare similar products from different suppliers and allocate orders among different products or services. This work focuses on allocating orders among multiple suppliers considering rebate. The objective function is to minimize the total acquisition cost including purchasing cost and rebate benefit. Rebate benefit is complex and difficult to estimate at the ordering step. Rebate rules vary for different suppliers and usually change over time. In this work, we developed a system to collect the rebate policies, standardized the rebate policies and developed two-stage optimization models for ordering allocation. Rebate policy with multi-tiers is considered in modeling. The discontinuous cost function of rebate benefit is formulated for different scenarios. A piecewise linear function is used to approximate the discontinuous cost function of rebate benefit. And a Mixed Integer Programing (MIP) model is built for order allocation problem with multi-tier rebate. A case study is presented and it shows that our optimization model can reduce the total acquisition cost by considering rebate rules.

Keywords: discontinuous cost function, mixed integer programming, optimization, procurement, rebate

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Authors: David Muyise

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Mixed Integer Programming (MIP) is an approach that involves the optimization of a range of decision variables in order to minimize or maximize a particular objective function. The main objective of this study was to apply the MIP approach to optimize the deployment of waste plastic to fuel oil processing plants in Uganda. The processing plants are meant to reduce plastic pollution by pyrolyzing the waste plastic into a cleaner fuel that can be used to power diesel/paraffin engines, so as (1) to reduce the negative environmental impacts associated with plastic pollution and also (2) to curb down the energy gap by utilizing the fuel oil. A programming model was established and tested in two case study applications that are, small-scale applications in rural towns and large-scale deployment across major cities in the country. In order to design the supply chain, optimal decisions on the types of waste plastic to be processed, size, location and number of plants, and downstream fuel applications were concurrently made based on the payback period, investor requirements for capital cost and production cost of fuel and electricity. The model comprises qualitative data gathered from waste plastic pickers at landfills and potential investors, and quantitative data obtained from primary research. It was found out from the study that a distributed system is suitable for small rural towns, whereas a decentralized system is only suitable for big cities. Small towns of Kalagi, Mukono, Ishaka, and Jinja were found to be the ideal locations for the deployment of distributed processing systems, whereas Kampala, Mbarara, and Gulu cities were found to be the ideal locations initially utilize the decentralized pyrolysis technology system. We conclude that the model findings will be most important to investors, engineers, plant developers, and municipalities interested in waste plastic to fuel processing in Uganda and elsewhere in developing economy.

Keywords: mixed integer programming, fuel oil plants, optimisation of waste plastics, plastic pollution, pyrolyzing

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Authors: Abu Hashan Md Mashud, M. Sharif Uddin, Aminur Rahman Khan

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In this paper, we formulate a new problem as a linear programming and Integer Programming problem and maximize profit within the limited budget and limited resources based on the construction of a tea garden problem. It describes a new idea about how to optimize profit and focuses on the practical aspects of modeling and the challenges of providing a solution to a complex real life problem. Finally, a comparative study is carried out among Graphical method, Simplex method and Branch and bound method.

Keywords: integer programming, tea garden, graphical method, simplex method, branch and bound method

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Authors: N. F. Jamaluddin, N. A. H. Aizam

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This paper explains the educational timetabling problem, a type of scheduling problem that is considered as one of the most challenging problem in optimization and operational research. The university examination timetabling problem (UETP), which involves assigning a set number of exams into a set number of timeslots whilst fulfilling all required conditions, has been widely investigated. The limitation of available timeslots and resources with the increasing number of examinations are the main reasons in the difficulty of solving this problem. Dynamical change in the examination scheduling system adds up the complication particularly in coping up with the demand and new requirements by the communities. Our objective is to investigate these demands and requirements with subjects taken from Universiti Malaysia Terengganu (UMT), through questionnaires. Integer linear programming model which reflects the preferences obtained to produce an effective examination timetabling was formed.

Keywords: demands, educational timetabling, integer linear programming, scheduling, university examination timetabling problem (UETP)

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Authors: Younghoon Kim, Seoung Bum Kim

Abstract:

One-class classification plays an important role in detecting outlier and abnormality from normal observations. In the previous research, several attempts were made to extend the scope of application of the one-class classification techniques to statistical process control problems. For most previous approaches, such as support vector data description (SVDD) control chart, the design of the control limits is commonly based on the assumption that the proportion of abnormal observations is approximately equal to an expected Type I error rate in Phase I process. Because of the limitation of the one-class classification techniques based on convex optimization, we cannot make the proportion of abnormal observations exactly equal to expected Type I error rate: controlling Type I error rate requires to optimize constraints with integer decision variables, but convex optimization cannot satisfy the requirement. This limitation would be undesirable in theoretical and practical perspective to construct effective control charts. In this work, to address the limitation of previous approaches, we propose the one-class classification algorithm based on the mixed integer programming technique, which can solve problems formulated with continuous and integer decision variables. The proposed method minimizes the radius of a spherically shaped boundary subject to the number of normal data to be equal to a constant value specified by users. By modifying this constant value, users can exactly control the proportion of normal data described by the spherically shaped boundary. Thus, the proportion of abnormal observations can be made theoretically equal to an expected Type I error rate in Phase I process. Moreover, analogous to SVDD, the boundary can be made to describe complex structures by using some kernel functions. New multivariate control chart applying the effectiveness of the algorithm is proposed. This chart uses a monitoring statistic to characterize the degree of being an abnormal point as obtained through the proposed one-class classification. The control limit of the proposed chart is established by the radius of the boundary. The usefulness of the proposed method was demonstrated through experiments with simulated and real process data from a thin film transistor-liquid crystal display.

Keywords: control chart, mixed integer programming, one-class classification, support vector data description

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Authors: Birce Boga Bakirli

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In this paper, we investigate ideas on human capital development and management in industries. We structured a model to be able to gather the data from the interviews conducted with worker, specialists and owners of companies. Different aspects of the situation are found in these interviews, and we used the information to model the benefit of the business owners and workers perspectives. These are modelled as a bi-level programming problem. Several instances of the generic cases are solved. The results show the importance of education within and out of the company for workers, and it returns for the company.

Keywords: bi-level programming, corporate strategy, cost tradeoffs, human capital, mixed integer programming, Stackelberg game, supplier relations, strategic planning

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Authors: Mohammad Pourmohammadi Fallah, Maziar Salahi

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In the literature, a mixed-integer linear programming model for the integrated production planning and warehouse layout problem is proposed. To solve the model, the authors proposed a Lagrangian relax-and-fix heuristic that takes a significant amount of time to stop with gaps above 5$\%$ for large-scale instances. Here, we present two heuristic algorithms to solve the problem. In the first one, we use a greedy approach by allocating warehouse locations with less reservation costs and also less transportation costs from the production area to locations and from locations to the output point to items with higher demands. Then a smaller model is solved. In the second heuristic, first, we sort items in descending order according to the fraction of the sum of the demands for that item in the time horizon plus the maximum demand for that item in the time horizon and the sum of all its demands in the time horizon. Then we categorize the sorted items into groups of 3, 4, or 5 and solve a small-scale optimization problem for each group, hoping to improve the solution of the first heuristic. Our preliminary numerical results show the effectiveness of the proposed heuristics.

Keywords: capacitated lot-sizing, warehouse layout, mixed-integer linear programming, heuristics algorithm

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Authors: Sangchul Kim, Dosaeng Park

Abstract:

It is one of the important design issues to provide various levels of difficulty in order to suit the skillfulness of an individual. In this paper we propose an integer programming-based method for selecting a mixture of challenges for a racing game that meet a given degree of difficulty. The proposed method can also be used to dynamically adjust the difficulty of the game during the progression of playing. By experiments, it is shown that our method performs well enough to generate games with various degrees of difficulty that match the perception of players.

Keywords: level generation, level adjustment, racing game, ip

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