Mixed Binary Integer Programming Models Research Articles

Bi-objective unrelated parallel machine joint scheduling of jobs and preventive maintenance with a dynamic speed-scaling technique

Nowadays, industries contribute to a substantial consumption of energy. This consumption leads to significant greenhouse gas emissions, energy dependence and ever-increasing costs for manufacturers. Controlling consumption and improving energy efficiency in industry are therefore major challenges in terms of environmental issues. In this context, this article examines an energy-efficient bi-objective unrelated parallel machine joint scheduling of jobs and preventive maintenance activities problem to minimize both makespan and total energy consumption. The parallel machines are speed-scaling. To solve the problem, we propose a bi-objective mixed integer linear programming (MILP) model. Since the problem involves assigning jobs to machines and selecting an appropriate processing speed level for each job, we characterize each individual by two vectors: a job-machine assignment vector and a speed vector. To the best of our knowledge, no papers considering both production scheduling and Preventive Maintenance periods with minimizing the bi-objective makespan and the total of energy consumption. The gap between theory and practice prevents the satisfying performance of the schedules in industry. The purpose of this paper is to establish a joint model for integrating the production and PM scheduling to optimize the bi-objective of the total of energy consumption (TEC) and makespan (Cmax) simultaneously; and to make the theory applied in practice more efficaciously. The performance of the proposed mixed binary integer programming model is evaluated based on the exact method of Branch and Bound integrated on the CPLEX solver plus analyses using MATLAB software. A study of the results proved the performance of the model developed.

Optimizing the supply of essential goods under closed-off management: A case study of COVID-19

The COVID-19 pandemic is threatening people's health and economic development around the world. Many areas and cities are opting for actions such as lockdown or closed-off management to prevent rapid spread of the virus. Under these actions, it is necessary and important to guarantee the supply of essential goods. In this paper, we develop a multi-objective optimization model to optimize the supply of essential goods under closed-off management. A mixed binary integer programming model is built to help identify the optimal safety stock level as well as the supply network so that the demands for essential goods could be guaranteed. The effectiveness of our method is demonstrated by an example. We also perform sensitivity analysis for key parameters. Our work provides a reasonable solution for the supply of essential goods under closed-off management.

Green Scheduling of Jobs and Flexible Periods of Maintenance in a Two-Machine Flowshop to Minimize Makespan, a Measure of Service Level and Total Energy Consumption

The success of an industry today depends on its ability to innovate. In terms of energy performance, this innovation is reflected in the ability of manufacturers to implement new solutions or technologies that enable better energy management. In this regard, this paper aims to address this gap by incorporating energy consumption as an explicit criterion in flowshop scheduling of jobs and flexible preventive maintenance. Leveraging the variable speed of machining operations leading to different energy consumption levels, we explore the potential for energy saving in manufacturing. We develop a mixed integer linear multiobjective optimization model for minimizing the makespan and the total energy consumption. In the literature, no papers considering both production scheduling and flexible periods of maintenance with minimizing both objective the total of energy consumption in flowshop and makespan. The performance of the proposed mixed binary integer programming model is evaluated based on the exact method of branch and bound algorithm. A study of the results proved the performance of the model developed.

Speed scaling technique integrated in scheduling of production and maintenance under energy constraints using genetic algorithms

Recently, energy scarcity and environmental pollution have become increasingly serious problems, while the scheduling of energy savings is attracting increasing attention from industries to minimize energy consumption. This article deals with the problem of scheduling production and maintenance under energy constraints in the flow shop. Two mixed binary integer programming models are provided to derive optimal scheduling to minimize total energy consumption (TEC). To solve this problem, we use an approximate method as a genetic algorithm (GA).

Scheduling Jobs on a Single Machine With Dirt Cleaning Consideration to Minimize Total Completion Time

This paper studies a single-machine scheduling problem observed in the wafer manufacturing process, where the machine must receive periodical maintenance so that the dirt generated in the process does not exceed the limit. The objective is to minimize the total completion times. A mixed binary integer programming model is formulated, and, due to its computational intractability for large problems, three effective heuristics are proposed based on our developed properties. The proposed heuristics are evaluated by comparison with exact solutions on small problems and with lower bounds on large problems. The experimental results show that the INDEX-LOE heuristic yields high-quality solutions in comparison with those obtained from the other two heuristics. Furthermore, the impacts of dirt accumulation and cleaning time are discussed in detail.

A Bi-Level Programming Model of Liquefied Petroleum Gas Transportation Operation for Urban Road Network by Period-Security

As a clean energy, Liquefied Petroleum Gas (LPG) is consistent with the coordinated and sustainable development of both the economy and environment. However, LPG is a hazardous material (hazmat) and is thus always transported in cylinders by vehicles on urban road networks to meet varying demand. This transport can threaten the surrounding citizens, vehicles, and even the whole urban area. Hence, LPG transportation should be focused on maintaining its security while simultaneously minimizing shipping costs. When LPG is moved through an urban area, its threat level fluctuates with the network congestion level, which continually varies by different time periods. So, variation in the magnitude of the threat posed by LPG transportation causes additional changes in the safe-related cost as well as the shipping cost. This study aims to solve the problem of an LPG transportation operation on an urban road network according to congested periods; the solution is based on cutting its two types of cost. In general, we should choose an LPG transport period that results in a lower safety cost, however optimization of an LPG transportation operation must minimize both the safety cost and shipping cost. This paper presents the problem of LPG flow distribution and vehicle dispatch scheme by “period-security” to rationalize the LPG transport risk level. Firstly, the impedance function of LPG flow distribution was constructed with a focus on the safety cost in different periods. Meanwhile, a bi-level programming model was built, in which the upper mixed binary integer programming model aims to minimize the shipping cost and the lower model is a user-equilibrium model that is aimed at calculating the distribution of the LPG demands on the given lines and in feasible periods. Then, we designed a heuristic algorithm based on the Genetic Algorithm to solve the upper model and embedded the Frank-Wolfe Algorithm to get the optimal LPG flow distribution solution. Numerical examples are presented which validate that the LPG optimal operation can realize a minimal safety cost and the minimum shipping cost for three LPG demand values by considering the congestion situation.

Minimizing the number of tardy jobs on unrelated parallel machines with dirt consideration

ABSTRACTThis paper considers an unrelated parallel machine scheduling problem with the objective of minimizing the number of tardy jobs. Each machine should stop periodically to perform maintenance activities. The problem, motivated from a wafer manufacturing company, considers the job scheduling and maintenance activities simultaneously under dirt constraint. That is, the dirt accumulation in the machine does not exceed the prespecified dirt limit. A mixed binary integer programming (MBIP) model is developed to find optimal solutions, and two three-phase heuristics are proposed. The heuristics assign each job to its most efficient machine first. Then, an intension of Moore’s algorithm is applied for each machine, and finally the solution is improved by the forward/backward insert mechanism. The experimental results showed that the proposed heuristics perform well. Furthermore, the efficiency of the MBIP model and the impact of the dirt accumulation as well as maintenance time are studied in detail.

Minimizing total absolute deviation of job completion times on a single machine with cleaning activities

This paper considers a new single machine scheduling problem with several unavailability periods, where the machine has to be stopped in order to remove the dirt. The problem appears in the process in wafer manufacturing company. During the processing of a wafer on a specific machine, the dirt, including particle, organic materials and metal-salts, etc., on the surface of the wafer while processing, has to be taken away by a cleaning agent. Once the accumulation of dirt is over a threshold value, it is necessary to interrupt the machine processing and replace the cleaning agent in order to avoid damaging the wafer. The objective is to minimize the total absolute deviation of job completion times (TADC). The problem integrates production scheduling and cleaning activities and is strongly NP-hard. Many properties for solving the problem efficiently are explored. A mixed binary integer programming model is developed to find the optimal solution. Based on the properties explored, symmetry of the V-shaped rule, and the dynamic programming on scheduling the cleaning activities, an effective heuristic is developed. Computational results indicate that the performance of the heuristic is robust and significantly outperforms the modified TADC solution in the literature. Furthermore, the efficiency of the mixed binary integer programming model and the impact of the dirt accumulation as well as cleaning time are explored in detail.

A hybrid genetic algorithm for non-permutation flow shop scheduling problems with unavailability constraints

This article deals with the non-permutation flow shop scheduling problems with non-availability intervals. Specifically, two kinds of unavailability constraints are investigated and the jobs are non-resumable in both cases. In the first case, the non-availability intervals are periodically fixed and known in advance. In the second case, the intervals are flexible and the machines’ continuous working time cannot exceed a maximum allowed time. Two mixed binary integer programming models are provided for deriving the optimal schedules, respectively. The problems of minimising makespan in such flow shops are proved Non-deterministic Polynomial-time hard in strong sense. Then a hybrid incremental genetic algorithm (HIGA) that combines an incremental evolution strategy framework, a local refinement and a population diversity supervision scheme is proposed to solve the large-sized problems efficiently. The numerical experiments under different problem parameters’ settings indicate that the HIGA can achieve quite satisfactory performance compared with genetic algorithm and a constructive heuristic based on Nawaz, Enscore, Ham.

Two metaheuristics to solve a multi-item multiperiod inventory control problem under storage constraint and discounts

In this paper, a multi-item multiperiod inventory control problem with all-unit and/or incremental quantity discount policies under limited storage capacity is presented. The independent random demand rates of the items in the periods are known and the items are supplied in distinct batch sizes. The cost consists of ordering, holding, and purchasing. The objective is to find the optimal order quantities of all items in different periods such that the total inventory cost is minimized and the constraint is satisfied. A mixed binary integer programming model is first developed to model the problem. Then, a parameter-tuned genetic algorithm (GA) is employed to solve it. Since there is no benchmark available in the literature, a memetic algorithm (MA) is utilized as well to validate and verify the results obtained. The model implementation is next presented using some numerical examples and finally the performances of the proposed GA and MA are compared using two statistical tests and a simple additive weighting method. The results show that GA has better performance than MA in terms of average objective function value and average run time using the two comparison procedures.

Optimizing blocking flow shop scheduling problem with total completion time criterion

The blocking flow shop scheduling problem has found many applications in manufacturing systems. There are a few exact methods for solving this problem with different criteria. In this paper, efforts will be made to optimize the total completion time criterion for this problem. We present two mixed binary integer programming models, one of which is based on the departure times of jobs from machines, and the other is based on the idle and blocking times of jobs. An initial upper bound generator and some lower bounds and dominance rules are also developed to be used in a branch and bound algorithm. The algorithm solves 17 instances of the Taillard's benchmark problem set in less than 20min.

Mixed Binary Integer Programming Formulation for the Two-Echelon Supply Chain System with Two Customer Locations

Coordinating works between production and delivery is a crucial issue for the management of production and logistics. This study developed a single machine scheduling model that incorporated delivery vehicle routing decisions and serves two customer locations. The objective was to minimize the mean arrival time. The problem is NP-hard in the strong sense in general. A mixed binary integer programming (BIP) model was developed to optimally solve this problem. In addition, computational results of the illustrative example are reported using the mixed BIP model to solve the proposed problem.

A coevolutionary algorithm for the flexible delivery and pickup problem with time windows

Abstract This paper addresses a flexible delivery and pickup problem with time windows (FDPPTW) and formulates the problem into a mixed binary integer programming model in order to minimize the number of vehicles and to minimize the total traveling distance. This problem is shown to be NP-hard. In this study, therefore, a coevolutionary algorithm incorporated with a variant of the cheapest insertion method is developed to speed up the solution procedure. The FDPPTW scheme overcomes the shortcomings of the existing schemes for the delivery and pickup problems. By testing with some revised Solomon's benchmark problems, the computational results have shown the efficiency and the effectiveness of the developed algorithm.

A genetic algorithm for the simultaneous delivery and pickup problems with time window

This paper concerns a Simultaneous Delivery and Pickup Problem with Time Windows (SDPPTW). A mixed binary integer programming model was developed for the problem and was validated. Due to its NP nature, a co-evolution genetic algorithm with variants of the cheapest insertion method was proposed to speed up the solution procedure. Since there were no existing benchmarks, this study generated some test problems which revised from the well-known Solomon’s benchmark for Vehicle Routing Problem with Time Windows (VRPTW). From the comparison with the results of Cplex software and the basic genetic algorithm, the proposed algorithm showed that it can provide better solutions within a comparatively shorter period of time.

Scheduling multiple orders per job in a single machine to minimize total completion time

This paper deals with a single-machine scheduling problem with multiple orders per job (MOJ) considerations. Both lot processing machines and item processing machines are also examined. There are two primary decisions that must be made in the proposed problem: (1) how to group the orders together, and (2) how to schedule the jobs once they are formed. In order to obtain the optimal solution to a scheduling problem, these two decisions should be made simultaneously. The performance measure is the total completion time of all orders. Two mixed binary integer programming models are developed to optimally solve this problem. Also, two efficient heuristics are proposed for solving large-sized problems. Computational results are provided to demonstrate the efficiency of the models and the effectiveness of the heuristics.

Minimizing maximum tardiness in single FMS machine scheduling with tool changes

Flexible Manufacturing System (FMS) is a programmable automation system that efficiently accommodates product variations and small-lot production capability. Tool change is an important issue for FMS tool management. Previous literature on tool management addresses this issue, but assumes that the change is due only to part mix. In practice, however, most tool changes are the result of tool wear. This study considers the problem of scheduling a set of jobs on a single FMS where the cutting tool is subject to wear, aiming to minimize the maximum tardiness. Two mixed binary integer programming models were developed to solve this problem optimally. We developed the heuristic method to compute result to demonstrate the efficiency of the models.

Scheduling of nonresumable jobs and flexible maintenance activities on a single machine to minimize makespan

Abstract This study addresses a single machine scheduling problem with periodic maintenance, where the machine is assumed to be stopped periodically for maintenance for a constant time w during the scheduling period. Meanwhile, the maintenance period [ u , v ] is assumed to have been previously arranged and the time w is assumed not to exceed the available maintenance period [ u , v ] (i.e. w ⩽ v − u ). The time u ( v ) is the earliest (latest) time at which the machine starts (stops) its maintenance. The objective is to minimize the makespan. Two mixed binary integer programming (BIP) models are provided for deriving the optimal solution. Additionally, an efficient heuristic is proposed for finding the near-optimal solution for large-sized problems. Finally, computational results are provided to demonstrate the efficiency of the models and the effectiveness of the heuristics. The mixed BIP model can optimally solve up to 100-job instances, while the average percentage error of the heuristic is below 1%.

A framework for an integrated distribution system optimisation model

This paper presents a framework for an integrated optimisation model of supply chain functions in a multi-plant, multi-product, multi-customer supply chain with deterministic demands. A Mixed-Binary Integer Programming model is proposed to integrate the Capacitated Location, Production and Distribution (CLPD) functions within a supply chain. The integrated CLPD problem is formulated as a profit maximisation model with capacity and demand constraints. The result is a framework from which to design entire supply chains from suppliers to customers. The proposed model is solved using OPL Studio 3.7 and solutions for four variations of the CLPD problem are presented to demonstrate the efficacy of the integrated approach. The main contribution of this research is a multi echelon, supply chain design and modelling hierarchy. This study was motivated by a perceived gap in the literature on supply chain modelling methods and supply chain frameworks.

Mixed binary integer programming formulations for the flow shop scheduling problems. A case study: ISD projects scheduling

In this paper, we consider a flow shop scheduling problem under different conditions. The model is formulated in a form of mixed binary integer programming. The primary goal of this paper is to develop operation schedules that specify the completion time of operations. The objective function of our model is formulated as a minimization of the makespan, the weighted mean flow-time or the weighted mean tardiness. There are several constraints which are involved in our modeling such as the due dates of jobs, the jobs ready times, etc. For these objective functions and constraints, seven mixed binary integer programming models are proposed. We demonstrate the implementation of our proposed models using different numerical examples. As a case study, we apply the proposed models for an information systems development (ISD) center to schedule the ISD projects using some real data.

Integer programming models for the re-entrant shop scheduling problems

A re-entrant shop describes a manufacturing environment in which a machine can process a job more than once. A re-entrant shop can be classified into a re-entrant job-shop (RJS) and a re-entrant flow-shop (RFS). This article presents the development of the mixed binary integer programming (BIP) technique for the RJS and the RFS scheduling problems. Eight mixed BIP models are proposed for solving re-entrant shop scheduling problems to minimize makespan. The models RJS-1, RJS-2, RJS-3, and RJS-4 are for the RJS problems, while the other four models (RFS-1, RFS-2, RFS-3, and RFS-4) are for the RFS problems. Computational results show that the RJS-4 model is the best of the RJS models, and the RFS-3 model is the best of the RFS models.