Proposed Mixed Integer Linear Programming Model Research Articles

Mathematical model and discrete artificial Bee Colony algorithm for distributed integrated process planning and scheduling

With the development of the globalization of economy and manufacturing industry, distributed manufacturing mode has become a hot topic in current production research. In the context of distributed manufacturing, one job has different process routes in different workshops because of heterogeneous manufacturing resources and manufacturing environments in each factory. Considering the heterogeneous process planning problems and shop scheduling problems simultaneously can take advantage of the characteristics of distributed factories to finish the processing task well. Thus, a novel network-based mixed-integer linear programming (MILP) model is established for distributed integrated process planning and scheduling problem (DIPPS). The paper designs a new encoding method based on the process network and its OR-nodes, and then proposes a discrete artificial bee colony algorithm (DABC) to solve the DIPPS problem. The proposed DABC can guarantee the feasibility of individuals via specially-designed mapping and switching operations, so that the process precedence constraints contained by the network graph can be satisfied in the entire procedure of the DABC algorithm. Finally, the proposed MILP model is verified and the proposed DABC is tested through some open benchmarks. By comparing with other powerful reported algorithms and obtaining new better solutions, the experiment results prove the effectiveness of the proposed model and DABC algorithm successfully.

Optimizing the workload of production units of a make-to-order manufacturing system

This paper proposes a production planning approach for a job shop type manufacturing company in electronics sector that operates with make-to-order (MTO) convention and has a high-mix production range. A novel Mixed Integer Linear Programming (MILP) model is proposed that finds workload-dependent planning horizon by making order acceptance decisions. Partial acceptance of orders and delayed internal target deliveries contribute to the novelty of the proposed model. We establish the complexity of the problem and solve real-life problem instances of a medium-size manufacturing company in the south of The Netherlands. Our computational results show that the proposed MILP model can allocate desired workload to work centers by dynamically adjusting the time horizon for production plans. Our approach can practically be used within reasonable times for daily updates of internal production targets. We perform a sensitivity analysis to show how the time horizon is determined with varying system conditions. The obtained workload amounts satisfy minimum requirements of production units, as this cannot be achieved by the current practice.

A Two-Level Simulation-Assisted Sequential Distribution System Restoration Model With Frequency Dynamics Constraints

This paper proposes a service restoration model for unbalanced distribution systems and inverter-dominated microgrids (MGs), in which frequency dynamics constraints are developed to optimize the amount of load restoration and guarantee the dynamic performance of system frequency response during the restoration process. After extreme events, the damaged distribution systems can be sectionalized into several isolated MGs to restore critical loads and tripped non-black start distributed generations (DGs) by black start DGs. However, the high penetration of inverter-based DGs reduces the system inertia, which results in low-inertia issues and large frequency fluctuation during the restoration process. To address this challenge, we propose a two-level simulation-assisted sequential service restoration model, which includes a mixed integer linear programming (MILP)-based optimization model and a transient simulation model. The proposed MILP model explicitly incorporates the frequency response into constraints, by interfacing with transient simulation of inverter-dominated MGs. Numerical results on a modified IEEE 123-bus system have validated that the frequency dynamic performance of the proposed service restoration model are indeed improved.

A MILP Model for Water Level Sensor Placement with Multi-Sensor and Multi-Disaster Areas

A water level sensor is critical to measure water levels at strategic points in the river. The sensor location directly impacts the quality of the collected data sent to the flood early warning system. To prevent and minimize the risk of flooding, it is crucial to determine the optimal locations for water level sensor placement. This research proposes a Mixed-Integer Linear Programming (MILP) model for water level sensor placement considering multi-sensor and multi-disaster areas. In addition, this model was applied in a case study in Tikala River, Manado, Indonesia. The results indicated that all disaster areas could be covered by at least one single sensor. A sensitivity analysis was performed by running the model under several different budget scenarios. When the budget increases, the number of sensors and the coverage performance are getting larger. Thus, the proposed MILP model was able to determine the optimal locations for sensor placement under a limited budget.

Scheduling on flexible flow shop with cost-related objective function considering outsourcing options

This study considers outsourcing decisions in a flexible flow shop scheduling problem, in which each job can be processed by either an in-house production line or outsourced. The selected objective function aims to minimize the weighted sum of tardiness costs, in-house production costs, and outsourcing costs with respect to the jobs due date. The purpose of the problem is to select the jobs that must be processed in-house, schedule processing of the jobs in-house, and finally select and assign other jobs to the subcontractors. We develop a mixed-integer linear programming (MILP) model for the research problem. Regarding the complexity of the research problem, the MILP model cannot be used for large-scale problems. Therefore, four metaheuristic algorithms, including SA, GA, PSO, hybrid PSO-SA, are proposed to solve the problem. Furthermore, some random test problems with different sizes are generated to evaluate the effectiveness of the proposed MILP model and solution approaches. The obtained results demonstrate that the GA can obtain better solutions in comparison to the other algorithms.

A robust MILP and gene expression programming based on heuristic rules for mixed-model multi-manned assembly line balancing

Current dynamic markets require manufacturing industries to organize a robust plan to cope with uncertain demand planning. This work addresses the mixed-model multi-manned assembly line balancing under uncertain demand and aims to optimize the assembly line configuration by a robust mixed-integer linear programming (MILP) model and a robust solution generation mechanism embedded with dispatching rules. The proposed model relaxes the cycle time constraint and designs robust sequencing constraints and objective functions to ensure the line configuration can meet all the demand plans. Furthermore, two solution generation mechanisms, including a task-operator-sequence and an operator-task-sequence, are designed. To quickly find a suitable line configuration, a gene expression programming (GEP) approach with multi-attribute representation is proposed to obtain efficient dispatching rules which are ultimately embedded into the solution generation mechanisms. Experimental results show that solving the proposed MILP model mathematically is effective when tackling small and medium-scale instances. However, for large instances, the dispatching rules generated by the GEP have significant superiority over traditional heuristic rules and those rules mined by a genetic programming algorithm.

Optimization of multiple-crane service schedules in overlapping areas through consideration of transportation efficiency and operational safety

Tower crane scheduling is a classic conundrum. It is further complicated to prevent collisions and reduce idle transportation time among multiple overlapping tower cranes. Unlike previous research attempts, this study aims to provide an optimal solution to this multiple crane service scheduling problem (MCSSP). Firstly, the MCSSP was translated to a Mixed Integer Linear Programming (MILP) model. Then, the model was optimized by 1) distributing the lifting requests in overlapping areas to the proper tower cranes; 2) selecting the appropriate supply location to serve each lifting request; and 3) arranging the lifting sequences of each tower crane to complete the requests. Compared with previous methods, the proposed MILP model (solved using Gurobi™) can result in the saving of 6.54%–18.07% of total operation costs meanwhile achieving the non-collision goal. The findings of this research can be deployed in optimizing efficiency and safety in the real-life scheduling of multiple overlapping tower cranes.

An integrated fleet planning model with empty vehicle repositioning for an intermodal transportation system

Fleet planning problems in intermodal transportation represent one of the most important problems in logistics. In addition to the integration of various sub-problems interrelated at different planning levels, i.e., strategic, tactical, and operational, the inclusion of multiple modes, resources, and multiple actors in intermodal transportation systems makes fleet planning more complex in comparison with unimodal systems. The present paper proposes a holistic fleet planning approach, which integrates various decisions such as freight planning (transport mode, service type and route selection), fleet composition and allocation, empty vehicle repositions, fleet expansion/reduction and outsourcing. A comprehensive mixed-integer linear programming (MILP) model is developed for an international logistics company, which operates a large intermodal network in Europe. The computational results of the case study have shown that application of a reallocation strategy may provide cost reduction up to 10% for the logistics company. Considering the existing situation of the company, outsourcing can be reduced up to 30%, and fleet utilization can be increased by 5% by applying the proposed model. The extensive case study also shows that effective and efficient fleet plans can be generated through the proposed MILP model.

Optimized Reactive Power Capability of Wind Power Plants With Tap-Changing Transformers

With the recent advancements in power electronics for wind turbines (WTs) and increasing penetration of wind energy, wind power plants (WPP) have become necessary contributors of reactive power support for the bulk power system. Balancing reactive power support with individual WT operating requirements in a cost-effective manner is a challenge for WPP designers. In this paper, we present a methodology to optimize the WPP reactive power capability as seen from the point of common coupling (PCC), accounting for steady-state operating capabilities of the WPP equipment. Thus, the proposed methodology determines the configuration of the tap-changing transformers within the WPP that maximizes the amount of reactive power the WPP can either consume or inject to the network, considering uncertain levels of wind power generation and voltage magnitudes at the PCC. The optimized reactive power capability (ORPC) problem is initially formulated as a mixed-integer nonlinear programming (MINLP) model. Then, a set of efficient linearization techniques are used to obtain a mixed-integer linear programming (MILP) model that can be solved via off-the-shelf mathematical programming solvers. Results demonstrate that the proposed MILP model is a scalable, flexible and accurate method to maximize the reactive power capability of WPP.

Mixed-Integer Linear Programming Model and Heuristic for Short-Term Scheduling of Pressing Process in Multi-Layer Printed Circuit Board Manufacturing

The main stages of printed circuit board (PCB) manufacturing are the design, fabrication, assembly, and testing. This paper focuses on the scheduling of the pressing process, which is a part of the fabrication process of a multi-layer PCB and is a new application since it has never been investigated in the literature. A novel mixed-integer linear programming (MILP) formulation for short-term scheduling of the pressing process is presented. The objective function is to minimize the makespan of the overall process. Moreover, a three-phase-PCB-pressing heuristic (3P-PCB-PH) for short-term scheduling of the pressing process is also presented. To illustrate the proposed MILP model and 3P-PCB-PH, the test problems generated from the real data acquired from a PCB company are solved. The results show that the proposed MILP model can find an optimal schedule for all small- and medium-sized problems but can do so only for some large-sized problems using the CPLEX solver within a time limit of 2 h. However, the proposed 3P-PCB-PH could find an optimal schedule for all problems that the MILP could find using much less computational time. Furthermore, it can also quickly find a near-optimal schedule for other large-sized problems that the MILP could not solved optimally.

OSCAR: On Optimizing Resource Utilization in Live Video Streaming

Live video streaming traffic and related applications have experienced significant growth in recent years. However, this has been accompanied by some challenging issues, especially in terms of resource utilization. Although IP multicasting can be recognized as an efficient mechanism to cope with these challenges, it suffers from many problems. Applying software-defined networking (SDN) and network function virtualization (NFV) technologies enable researchers to cope with IP multicasting issues in novel ways. In this article, by leveraging the SDN concept, we introduce OSCAR (Optimizing reSourCe utilizAtion in live video stReaming) as a new cost-aware video streaming approach to provide advanced video coding (AVC)-based live streaming services in the network. In this article, we use two types of virtualized network functions (VNFs): virtual reverse proxy (VRP) and virtual transcoder function (VTF). At the edge of the network, VRPs are responsible for collecting clients’ requests and sending them to an SDN controller. Then, by executing a mixed-integer linear program (MILP), the SDN controller determines a group of optimal multicast trees for streaming the requested videos from an appropriate origin server to the VRPs. Moreover, to elevate the efficiency of resource allocation and meet the given end-to-end latency threshold, OSCAR delivers only the highest requested quality from the origin server to an optimal group of VTFs over a multicast tree. The selected VTFs then transcode the received video segments and transmit them to the requesting VRPs in a multicast fashion. To mitigate the time complexity of the proposed MILP model, we present a simple and efficient heuristic algorithm that determines a near-optimal solution in polynomial time. Using the MiniNet emulator, we evaluate the performance of OSCAR in various scenarios. The results show that OSCAR surpasses other SVC- and AVC-based multicast and unicast approaches in terms of cost and resource utilization.

Mathematical Analysis of Uncertainty in Bangladesh Agricultural Sector

This study presents three different mathematical models for profit optimization of agricultural products in Bangladesh. The prime focus of the paper has been to develop a Mixed Integer Linear Programming (MILP) model and analyze this model for two situation of demand uncertainty. Considering demand will be known before and after production. For the mentions of above two cases, we investigate the change of solution applying least demand, maximum perhaps demand and extreme demand scenarios. I think this is real life problem and this analysis will be helpful for all types of agricultural producers. The proposed MILP model is to maximize the total profit and also to estimate the profitable production locations. The formulated MILP model were solved by A Mathematical Programming Language (AMPL) and results obtained by appropriate solver MINOS. Numerical example with the sensitivity of several parameters has been deployed to validate the models. Results show that maximum perhaps demand scenario gets better solution according to our expected value compare of other two scenarios.

A coordinated scheduling problem for the supply chain in a flexible job shop machine environment

In this study, a new coordinated scheduling problem is proposed for the multi-stage supply chain network. A multi-product and multi-period supply chain structure has been developed, including a factory, warehouses, and customers. Furthermore, the flexible job shop scheduling problem is integrated into the manufacturing part of the supply chain network to make the structure more comprehensive. In the proposed problem, each product includes a sequence of operations and is processed on a set of multi-functional machines at the factory to produce the final product. Final products are delivered to the warehouses to meet customers’ demands. If the demands of customers are not fulfilled, the shortage in the form of backorder may occur at any period. The problem is expressed as a bi-objective mixed-integer linear programming (MILP) model. The first objective function is to minimize the total supply chain costs. On the other hand, the second objective function aims to minimize the makespan in all periods. A numerical example is presented to evaluate the performance of the proposed MILP model. Five multi-objective decision-making (MODM) methods, namely weighted sum, goal programming, goal attainment, LP metric, and max–min, are used to provide different alternative solutions to the decision-makers. The performance of the methods is evaluated according to both objective function values and CPU time criteria. In order to select the best solution technique, the displaced ideal solution method is applied. The results reveal that the weighted sum method is the best among all MODM methods.

Optimization of Flow Shop Scheduling in Precast Concrete Component Production via Mixed‐Integer Linear Programming

The increasing number of prefabrication projects has increased the demand for precast concrete (PC) components. The production cost of PC components significantly affects the development of the precast industry and the progress of prefabrication projects. To reduce the production cost, both the delivery delay time and component storage time must be reduced. Flow‐arrangement optimization is generally performed using the genetic algorithm. However, this method cannot always yield a perfect optimal solution. Moreover, the traditional optimization model does not consider the impact of the overtime hours of workers on the project costs. In this study, a mixed‐integer linear programming (MILP) model was developed to optimize the production scheduling by minimizing the storage and delay times. The total delay time for the components was reduced by 55.3%, from 3.8 to 1.7 h, and the total storage time for finished components was reduced by 20.3%, from 6.4 to 5.1 h. Then, the use of the MILP model was extended to optimize the production scheduling by minimizing overtime. Finally, the feasibility and effectiveness of MILP were verified by comparing the results. The total overtime decreased by approximately 24.5%, from 11.5 to 9.3 h. It has been demonstrated that the proposed MILP model can achieve a better production sequence with less overtime. The findings of this research can be deployed in optimizing efficiency in the real‐life scheduling of production sequence.

Propagation-Based Network Partitioning Strategies for Parallel Power System Restoration With Variable Renewable Generation Resources

Grid resiliency is defined as the ability to prepare for and adapt to changing conditions and withstand and recover rapidly from disruptions. Following large-scale disruptions of the power grid such as a complete blackout, parallel power system restoration accelerates the recovery process by allowing for simultaneous restoration in each island through network partitioning, thus enhancing grid resiliency. However, existing network partitioning strategies have not sufficiently considered variable renewable resources yet, and may not be flexible or computationally efficient to accommodate additional requirements from the evolving bulk power grid. To bridge these gaps, this study proposes a propagation-based optimization approach for power grid network partitioning. The requirements of blackstart resources, power balancing, synchrocheck relays for ties lines, and increased ramp rate requirements due to variable renewable resources are modeled in the constraints. Through propagation on the physical connectivity of grid assets, the network partitioning issue is formulated as a mixed integer linear programming (MILP) problem. The linearity ensures highly computational performance and optimality of the solutions from the proposed approach to identify the cutset edge possibilities for bulk power grid applications. Graph reduction strategies are proposed to further improve the computational performance for online or nearly online applications. Case studies based on two IEEE benchmark systems show that the proposed MILP model is able to determine the optimal cutset for network partitioning in approximately 0.28 seconds, and the proposed network reduction strategy is able to further improve the computational performance by approximately 9%.

MILP model to decrease the number of workers in assembly lines with human-robot collaboration

The benefits of automatization in human-operated assembly lines are well known. When complete automatization is not possible, a combined production system, with both manual and automated parts, could be the best configuration. Adding robots to an only human-operated assembly line may decrease the number of stations and consequently the number of workers in the line. Even if, the number of stations cannot be decreased, the number of workers could be reduced in case of proper reallocation of tasks. In this paper, three mixed-integer linear programming (MILP) models are suggested to minimize in three steps the number of workers, the number of robots, and the cycle time in an assembly line with a predefined number of stations and partial automatization. The results for the optimization problems are shown based on the data of an assembly line producing power inverters. The proposed MILP model is implemented in the AIMMS modeling environment.

Data-driven robust strategies for joint optimization of rail renewal and maintenance planning

We study the problem of rail renewal and maintenance planning. The problem is to determine when and what type of maintenance tasks or rail renewal are required to be performed on different segments to maintain the rail in a safe and reliable condition. This problem is formulated as a Mixed Integer Linear Programming (MILP) model. The model applies Track Quality Index and also defines a new index to represent the current condition of the rail. Maintenance recovery effect is intrinsically uncertain; therefore, we develop data-driven uncertainty set approximation approaches and leverage robust optimization to handle the uncertainty. Data-driven uncertainty sets are constructed by approximating convex hulls of uncertain data points and by adding cutting planes to mix of classic robust uncertainty sets. We also obtained the robust counterpart formulations of the proposed MILP model for constructed uncertainty sets. Furthermore, a heuristic algorithm is developed to facilitate solving large-scale instances. Applicability and efficiency of the proposed approach are demonstrated through an illustrative case study of a Class I freight railroad network in the United States. Our analyses reveal that the proposed approaches introduce efficient strategies to deal with uncertainties in rail networks at the reasonable cost of increasing the budget.

An ACOPF-based bilevel optimization approach for vulnerability assessment of a power system

This paper examines the effects of reactive power dispatch, losses, and voltage profile on the results of the interdiction model to analyze the vulnerability of the power system. First, an attacker-defender Stackelberg game is introduced. The introduced game is modeled as a bilevel optimization problem where the attacker is modeled in the upper level and the defender is modeled in the lower level. The AC optimal power flow (ACOPF) is proposed as the defender’s tool in the lower-level problem to mitigate the attack consequences. Our proposed ACOPF-based mathematical framework is inherently a mixed-integer bilevel nonlinear program (MIBNLP) that is NP-hard and computationally challenging. This paper linearizes and then transforms it into a one-level mixed-integer linear program (MILP) using the duality theory and some proposed linearization techniques. The proposed MILP model can be solved to the global optimum using state-of-the-art solvers such as Cplex. Numerical results on two IEEE systems and Iran’s 400-kV transmission network demonstrate the performance of the proposed MILP for vulnerability assessment. We have also compared our MILP model with the DCOPF-based approach proposed in the relevant literature. The comparative results show that the reported damage measured in terms of load shedding for the DCOPF-based approach is always lower than or equal to that for the ACOPF-based approach and these models report a different set of critical lines, especially in more stressed and larger power systems. Also, the effectiveness and feasibility of the proposed MILP model for power-system vulnerability analysis are discussed and highlighted.

Industry 4.0 Model for circular economy and cleaner production

In today’s competitive scenario, the manufacturing industries are lagging behind in implementing the Industry 4.0 concept or in integrating smart, ubiquitous components due to high cost and high energy consumption due to the volatile market. The digital transformation has reshaped the manufacturing industries and has paved the way towards data-driven, intelligent, networked, and resilient manufacturing systems. In this context, Industry 4.0 is progressing exponentially and offers a productive output in terms of circular economy and cleaner production to attain ethical business by achieving accuracy, precision, and efficiency. Hence, there is a strong requirement to revamp the traditional manufacturing set-ups into smart manufacturing to gain self-adaptability, reliability, and flexibility with high quality and low-cost output. The paper proposes a mixed-integer linear programming (MILP) model for Industry 4.0 set-up to achieve circular economy and cleaner production by optimizing products-machine allocation. The proposed MILP model optimizes the trade-off between energy consumption and machine processing cost to gain a circular economy and cleaner production respectively. The proposed model also achieves ethical business by deploying sensors to capture real-time information to establish the Industry 4.0 facility. The paper discusses the product-machine specific analysis to optimize the manufacturing of customized and high-end products at low production cost with minimal energy consumption. The objective of the paper is to minimize the total cost and energy consumption of machines for establishing the Industry 4.0 facility to gain a circular economy and cleaner production. The proposed model is demonstrated and computationally tested on small, moderate, and large data instances and presented with their detailed analysis.

Formula omitted] Multi-contingency transient stability constrained AC optimal power flow with volt/var controllers

In order to guarantee the continuous operation of electric power systems (EPS), a proper planning of these networks in steady and transient state must be performed. This paper presents an N−1 multi-contingency AC optimal power flow (OPF) that embeds, in the same mathematical optimization model, a set of transient stability constraints (TSC) that guarantee rotor angle and angular velocity variation within technical limits, at given N−1 contingencies. Furthermore, the proposed model considers the operation of volt/var controllers, such as shunt elements and OLTC transformers, to further improve the operation of the EPS, under given levels of demand and generation. Taking advantage of the classical first-order transient stability model of synchronous machines and the implicit trapezoidal integration rule, the proposed model can be formulated as a stand-alone mixed-integer nonlinear programming (MINLP) model. Then, through well-established linearization techniques, the initially proposed MINLP model is transformed into a new mixed-integer linear programming (MILP) model, which can be implemented via algebraic programming languages, such as AMPL, and solved using convex optimization solvers, such as CPLEX. Three systems with dissimilar number of synchronous machines and nodes have been used for tests (i.e., the 9-Bus/3-Generator Western System Coordinating Council (WSCC) system, the 39-Bus/10-Generator New England system and the 68-Bus/16-Generator IEEE system). The efficiency of the proposed linearization techniques and the stability requirements of the solutions have been validated using an exact AC power flow and the transient stability analysis program PSAT-Matlab. Results show the ability of the proposed MILP model to provide stable operating points under N−1 contingencies, at minimum production cost.