Integer Mathematical Model Research Articles

A mixed integer programming model for a closed-loop supply-chain network

In this paper, a new mixed integer mathematical model for a closed-loop supply-chain network that includes both forward and reverse flows with multi-periods and multi-parts is proposed. The proposed model guarantees the optimal values of transportation amounts of manufactured and disassembled products in a closed-loop supply chain while determining the location of plants and retailers. Finally, computational results are presented for a number of scenarios to show and validate the applicability of the model.

A Mixed Integer Mathematical Programming Optimization Model for the NP-Hard Industrial Processes Scheduling Problem

This paper investigates single machine scheduling problem with unequal release times and deterioration jobs where the objective function of it is minimal makespan. A mixed integer mathematical programming optimization model is developed for the problem which belongs to the NP-hard problem. The model is tested on examples and compared with a heuristic algorithm introduced by Lee et al.. Moreover, the branch-bound algorithm proposed by Lee et al. also can obtained optimal solutions the same as the results from CPLEX but the CPU time of it for 28 jobs even needs more than 2 hours, thus, the results between them are not compared. According to results of computational tests, it is showed that the proposed model is very effective in solving problems because it can obtain optimal solutions within a short time. Therefore, it is very useful and valuable for decision maker who requires the optimal solutions.

A MULTI-OBJECTIVE APPROACH TO OPTIMIZE A PERIODIC MAINTENANCE POLICY

The present paper proposes a multi-objective approach to find out an optimal periodic maintenance policy for a repairable and stochastically deteriorating multi-component system over a finite time horizon. The tackled problem concerns the determination of the system elements to replace at each scheduled and periodical system inspection by ensuring the simultaneous minimization of both the expected total maintenance cost and the expected global system unavailability time. It is assumed that in the case of system elements failure they are instantaneously detected and repaired by means of minimal repair actions in order to rapidly restore the system. A nonlinear integer mathematical programming model is developed to solve the treated multi-objective problem, whereas the Pareto optimal frontier is described by the Lexicographic Goal Programming and the ε-constraint methods. To explain the whole procedure, a case study is solved and the related considerations are given.

Otimização do planejamento hierárquico da produção em usinas cooperadas do setor sucroenergético

Neste trabalho apresenta-se um modelo de otimização para o planejamento agregado da produção em usinas cooperadas do setor sucroenergético. Esta modelagem considera a relação hierárquica existente entre o planejamento anual da cooperativa e o planejamento tático de safra das usinas cooperadas. No nível de decisão da cooperativa, o modelo indica a meta de produção de cada usina e define a política de estocagem e de atendimento da demanda. No nível de decisão de uma usina cooperada, o modelo possibilita definir a quantidade de cana-de-açúcar colhida, a quantidade transportada por prestador de serviço, a seleção dos processos de produção de açúcar, álcool, melaço e energia elétrica. Para resolver os modelos de programação linear e programação inteira mista envolvidos, utilizou-se uma linguagem de modelagem algébrica e um software de última geração de programação matemática. Resultados computacionais obtidos em estudo de caso são apresentados como forma de ilustrar e validar a aplicação.

Possibilistic inventory and supplier selection model for an assembly system

In this paper, the problem of inventory lot-sizing and supplier selection for an assembly system is considered, where the supplier available capacities are assumed as ambiguous dynamic parameters. In this scenario, which is a frequent case in large assembly-based factories such as automobile manufacturers, the final product is assembled from multiple components with different conversion factors, which can be sourced from multi-capacitated suppliers through the multi-period horizon of imprecise demand. Due to high shut-down costs of assembly lines, it is assumed that production never stops even though some components may not be available. Therefore, the unfinished products are transferred to a buffer zone and preserved there until the lacking components become available. In this study, a possibilistic mixed integer mathematical model, with fuzzy objective function and soft constraints, is developed to determine which component in what quantities, from which suppliers, and in which periods should be ordered. The model, inspired by the real case of the Iran Khodro Car Company, aims to maximize the profit while keeping a high customer service level by avoiding shortages. This model also considers the ambiguity of dynamic parameters such as demand, suppliers’ available capacities, prices, and holding and shortage costs. To solve the problem, the possibilistic model is first converted into an auxiliary crisp multi-objective model. Through an interactive fuzzy approach, the suggested multi-objective problem is then transformed into an equivalent single-objective model. Finally, a particle swarm optimization is proposed to achieve the overall satisfactory compromise solution. A numerical sample is used to validate the proposed model.

Supply chain optimisation with U-type assembly line balancing

In supply chain optimisation problems, determining the location and number of facilities is considered at a strategic level, while mid-term and short-term decisions such as those concerning assembly policy, inventory levels, lot sizes and scheduling are handled at the tactical and operational levels. However, considering these decisions simultaneously is almost always ignored during the optimisation of distribution/production processes. The aim of the study is to optimise the supply chain network (strategic level), including manufacturers, assemblers and customers, while simultaneously balancing the U-type assembly lines (tactical level) in assemblers. A nonlinear mixed integer mathematical model is proposed to minimise the total costs and the number of assembly stations while minimising the total fixed costs of stations. To show the validity and usefulness of the proposed model, a numerical example with different scenarios and sensitivity analyses is given and discussed.

Optimizing multi-item multi-period inventory control system with discounted cash flow and inflation: Two calibrated meta-heuristic algorithms

A mixed binary integer mathematical programming model is developed in this paper for ordering items in multi-item multi-period inventory control systems, in which unit and incremental quantity discounts as well as interest and inflation factors are considered. Although the demand rates are assumed deterministic, they may vary in different periods. The situation considered for the problem at hand is similar to a seasonal inventory control model in which orders and sales happen in a given season. To make the model more realistic, three types of constraints including storage space, budget, and order quantity are simultaneously considered. The goal is to find optimal order quantities of the products so that the net present value of total system cost over a finite planning horizon is minimized. Since the model is NP-hard, a genetic algorithm (GA) is presented to solve the proposed mathematical problem. Further, since no benchmarks can be found in the literature to assess the performance of the proposed algorithm, a branch and bound and a simulated annealing (SA) algorithm are employed to solve the problem as well. In addition, to make the algorithms more effective, the Taguchi method is utilized to tune different parameters of GA and SA algorithms. At the end, some numerical examples are generated to analyze and to statistically and graphically compare the performances of the proposed solving algorithms.

Mathematical model and lower bounds for multi stage jobshop scheduling problem with special blocking constraint

This paper deals with multi stage jobshop scheduling problem. At each stage, either a fix or a variable number of identical parallel machines are available to execute jobs operations. As often in industrial environment, no space is available between machines, a specific blocking constraint is taken into account. A linear integer mathematical model is proposed and executed in XPress-MP optimization software to solve short size instances of this problem and to validate proposed model. In order to be able to evaluate future heuristics solutions quality, four lower bounds are proposed, computed and evaluated with benchmark for symmetric and asymmetric machines repartition cases.

Note on “A unique integer mathematical model for scheduling deteriorating jobs with rate-modifying activities on a single machine”

In this note, we study scheduling problems with simultaneous considerations of deteriorating jobs and rate-modifying activities on a single machine. The job-dependent deterioration effect and the job-independent deterioration effect models are examined with the objective of minimizing the makespan. We propose polynomial time solutions for all the studied problems.

A model for setting services on auxiliary bus lines under congestion

In this paper, a mathematical programming model and a heuristically derived solution is described to assist with the efficient planning of services for a set of auxiliary bus lines (a bus-bridging system) during disruptions of metro and rapid transit lines. The model can be considered static and takes into account the average flows of passengers over a given period of time (i.e., the peak morning traffic hour). Auxiliary bus services must accommodate very high demand levels, and the model presented is able to take into account the operation of a bus-bridging system under congested conditions. A general analysis of the congestion in public transportation lines is presented, and the results are applied to the design of a bus-bridging system. A nonlinear integer mathematical programming model and a suitable approximation of this model are then formulated. This approximated model can be solved by a heuristic procedure that has been shown to be computationally viable. The output of the model is as follows: (a) the number of bus units to assign to each of the candidate lines of the bus-bridging system; (b) the routes to be followed by users passengers of each of the origin–destination pairs; (c) the operational conditions of the components of the bus-bridging system, including the passenger load of each of the line segments, the degree of saturation of the bus stops relative to their bus input flows, the bus service times at bus stops and the passenger waiting times at bus stops. The model is able to take into account bounds with regard to the maximum number of passengers waiting at bus stops and the space available at bus stops for the queueing of bus units. This paper demonstrates the applicability of the model with two realistic test cases: a railway corridor in Madrid and a metro line in Barcelona.

A unique integer mathematical model for scheduling deteriorating jobs with rate-modifying activities on a single machine

In this paper, we consider the scheduling a set of deteriorating jobs on a single processor. We propose a model which reflects real life situations in which the processing time of jobs change depending on its initial processing time (deteriorating jobs) and activities, such as machine maintenance or a break for workers. A rate-modifying activity (rma) is a maintenance activity given to a machine to restore it to its original state. The general problem we deal with in this study is to determine the job sequence, the number of rmas, and rma positions within the work sequence. We formulate a unique integer program to solve this model for makespan and total completion time objectives. We also propose efficient heuristic algorithms for solving large size problems for both makespan and total completion time objectives.

Quay crane scheduling with time windows, one-way and spatial constraints

In this paper, we address the quay crane scheduling problem in maritime container terminals, with some new features on the machine environment. We consider rail-mounted cranes and we assume that they are not allowed to span all over the vessel and that they can work only in predefined time windows. We furthermore take into account one-way constraints on the crane movements. We present a mixed integer mathematical model and a heuristic algorithm to get feasible solutions to the problem. Finally, we report on some computational experience.

A graph theoretic-based heuristic algorithm for responsive supply chain network design with direct and indirect shipment

The configuration of the supply chain network has a strong influence on the overall performance of the supply chain. A well designed supply chain network provides a proper platform for efficient and effective supply chain management. The supply chain network should be designed in the way that could meet the customer needs with an efficient cost. This paper studies the responsive, multi-stage supply chain network design (SCND) problem under two conditions: (1) when direct shipment is allowed and (2) when direct shipment is prohibited. First, two mixed integer programming models are proposed for multi-stage, responsive SCND problem under two abovementioned conditions. Then, to escape from the complexity of mixed integer mathematical programming models, graph theoretic approach is used to study the structure of the SCND problems and it is proven that both of SCND problems considered in this paper could be modeled by a bipartite graph. Finally, since such network design problems belong to the class of NP-hard problems, a novel heuristic solution method is developed based on a new solution representation method derived from graph theoretic view to the structure of the studied problem. To assess the performance of the proposed heuristic solution method, the associated results are compared to the exact solutions obtained by a commercial.

The PHEV Charging Infrastructure Planning (PCIP) Problem

Increasing debates over a gasoline independent future and the reduction of greenhouse gas (GHG) emissions has led to a surge in plug-in hybrid electric vehicles (PHEVs) being developed around the world. The majority of PHEV related research has been directed at improving engine and battery operations, studying future PHEV impacts on the grid, and projecting future PHEV charging infrastructure requirements. Due to the limited all-electric range of PHEVs, a daytime PHEV charging infrastructure will be required for most PHEV daily usage. In this paper, for the first time, we present a mixed integer mathematical programming model to solve the PHEV charging infrastructure planning (PCIP) problem for organizations with thousands of people working within a defined geographic location and parking lots well suited to charging station installations. Our case study, based on the Oak Ridge National Laboratory (ORNL) campus, produced encouraging results, indicates the viability of the modeling approach and substantiates the importance of considering both employee convenience and appropriate grid connections in the PCIP problem.

Designing a mathematical model for dynamic cellular manufacturing systems considering production planning and worker assignment

Since workers have an important role in doing jobs on machines, assignment of workers to cells becomes a crucial factor for full utilization of cellular manufacturing systems. This paper presents an integer mathematical programming model for the design of cellular manufacturing systems in a dynamic environment. The advantages of the proposed model are as follows: considering multi-period production planning, dynamic system reconfiguration, duplicate machines, machine capacity, available time of workers, and worker assignment. The aim of the proposed model is to minimize holding and backorder costs, inter-cell material handling cost, machine and reconfiguration costs and hiring, firing and salary costs. Computational results are presented by solving some examples.

Global Access Network Evolution

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <?Pub Dtl=""?>In this paper, we propose to tackle the problem of updating the access network in order to connect new subscribers and to satisfy the new class of service requirements for the existing subscribers to offer, for instance, new services such as high-definition television (HDTV) over the Internet protocol (IPTV). Four important access network architectures/technologies are considered: the digital subscriber line (xDSL) technologies deployed directly from the central office (CO), the fiber-to-the-node (FTTN), the fiber-to-the-micro-node (FTTn) and the fiber-to-the-premises (FTTP). An integer mathematical programming model is proposed for this network planning problem. Next, a heuristic algorithm based on the tabu search principle is proposed to find “good” feasible solutions within a reasonable amount of computational time. Finally, numerical results are presented and analyzed. To assess the quality of the solutions found with the proposed algorithm, they are compared to the optimal solutions found using a commercial implementation of the branch-and-bound algorithm. </para>

The global solver in the LINDO API

The global solver in the LINDO Application Programming Interface (LINDO API) finds guaranteed global optima to nonconvex, nonlinear and integer mathematical models using the branch and bound/relax approach. We describe (a) the class of problems for which it tends to be appropriate; (b) how to access it directly via the LINDO API or via various modelling language front ends; (c) heuristics used for finding good initial solutions; (d) methods for constructing easily solved relaxations; (e) branching rules for splitting a problem into more easily solved subproblems; and (f) some illustrative computational results.

The integrated secondary route network design model in the hierarchical hub-and-spoke network for dual express services

The network industries design and operate physical distribution networks to provide services for spatial interactions. In the hierarchical hub-and-spoke network, each center is connected through a secondary route to its designated hub, while hubs are mutually connected by primary routes. It is a variation of pure hub-and-spoke networks that may eliminate center-hub partial loads with the result of higher load factor and lower cost. We study the integrated hierarchical hub-and-spoke network design problem for dual services. It integrates otherwise mutually exclusive secondary route networks for their respective services so that the total operating cost is minimized while meeting the service time and operations restrictions. Assuming symmetry in cost/time, it is equivalent to the integration of dual degree and time constrained trees. We propose a directed network configuration and formulate a link-based integer mathematical model. We also develop a link-based implicit enumeration with an embedded degree and time constrained spanning tree algorithm. The computational result showed that the integration provides a substantial cost reduction over the conventional exclusive secondary route networks for their respective services. In addition, the proper selection of number and location of hubs may achieve a higher cost-effectiveness as well as an increase in service territory at no additional operating cost.

Feature-based hybrid inspection planning: A mathematical programming approach

Intelligent planning for inspection of parts with complex geometric surfaces using contact or non-contact devices is still a major challenge. Contact measurement is widely used in manufacturing owing to its superiority in point accuracy. However, the volumetric accuracy of non-contact measurement techniques is better owing to the large number of points that can be measured in a short time. Consequently, contact measurement is usually used for mechanical parts with prismatic shapes while non-contact measurement methods are mostly used with free-form shapes. Complex parts that include both prismatic and free form shapes may require inspection using both techniques. It may not be possible to fully digitise a part using a single type of sensor owing to occlusion or accessibility issues. This paper proposes a hybrid (contact/non-contact) inspection planning approach that capitalises on the advantages of both inspection techniques. In the beginning, a knowledge-based system has been developed for selecting the most suitable sensor for the inspected features using a proposed inspection-specific features taxonomy. Additionally, a new travel salesperson problem (TSP) formulation has been developed for sequencing of hybrid inspection tasks, where a novel sub-tour elimination constraint has been formulated. The proposed 0–1 integer mathematical model minimises the non-digitisation related time between successive inspection operations. The developed hybrid inspection planning system not only overcomes the incompleteness of information when each sensor type is used separately, but also improves the accuracy of the point cloud obtained by using both sensors. The developed hybrid inspection planner was applied to the inspection of a water pump housing of an automotive engine. The applicability of the developed inspection planning framework and methodology for inspecting complex parts and associated dies and moulds has been demonstrated. The developed system makes it easier to utilise the powerful combination of measurement sensors while automating the development of an optimal sensor-task assignment as well as inspection tasks sequence. It has the potential benefits of reducing the inspection time and cost and increasing the efficiency of the whole inspection process, which have positive effects on the early product development stages and quality assurance activities.

A mixed integer programming model for remanufacturing in reverse logistics environment

Recently, there has been a growing interest in reverse logistics due to environmental deterioration. Firms incorporate reverse flow to their systems for such reasons as ecological and economic factors, government regulations and social responsibilities. In this paper a new mixed integer mathematical model for a remanufacturing system, which includes both forward and reverse flows, is proposed and illustrated on a numerical example. The proposed model provides the optimal values of production and transportation quantities of manufactured and remanufactured products while solving the location problem of dissassembly, collection and distribution facilities. The model is validated by using a set of experimental data reflecting practical business situation. Sensitivity analysis of the model is also presented.