mixed-integer linear programming

This paper addresses the open shop scheduling problem, considering parallel machines within each stage and integrating job transportation times between stages, independent of job specifics. In this scheduling problem, all jobs traverse each stage, and once a job commences on a machine, it must complete without machine breakdowns. To meet this challenge, a mixed-integer linear programming (MILP) model is introduced to minimize the makespan, which represents the maximum job completion time. Given the NP-hard nature of the open-shop scheduling problem, this study employs the whale metaheuristic algorithm to solve instances across various dimensions, spanning small, medium, and large scales. The algorithm parameters are systematically optimized using the Taguchi Method. Results from comparing the whale algorithm with the linear model implemented in GAMS highlight its exceptional efficiency in handling randomly generated small and medium-sized instances. Moreover, in a comparative analysis with other algorithms such as PSO and DE, the whale algorithm not only competes effectively but, in some instances, outperforms its counterparts. This observation underscores the algorithm's prowess in maintaining efficiency and high performance, particularly when addressing large-scale open-shop scheduling challenges. It excels in achieving a delicate balance between exploration and exploitation, thereby avoiding local optimal solutions.

This paper considers a customer order scheduling (COS) problem in which each customer requests a variety of products processed in a two-machine flow shop. A sequence-independent attached setup for each machine is needed before processing each product lot. We assume that customer orders are satisfied by the job-based processing approach in which the same products from different customer orders form a product lot (job). Each customer order for a product is processed as a sublot (a batch of identical items) of the product lot by applying the lot streaming (LS) idea in scheduling. We assume that all sublots of the same product must be processed together by the same machine without intermingling the sublots of other products. The completion time of a customer order is the completion time of the product processed as the last product in that order. All products in a customer order are delivered in a single shipment to the customer when the processing of all the products in that customer order is completed. We aim to find an optimal schedule with a product lots sequence and the sequence of the sublots in each job to minimize the sum of completion times of the customer orders. We have developed a mixed-integer linear programming (MILP) model and a multi-phase heuristic algorithm for solving the problem. The results of our computational experiments show that our model can solve the small-sized problem instances optimally. However, our heuristic algorithm finds optimal or near-optimal solutions for the medium- and large-sized problem instances in a short time.

Increasing software as a service (SaaS) requires the provision of more updated models for services, so trying to develop a model customized for the customer is important. We used the linear Knapsack problem model proposed by Mike Hewitt and Emma Frejinger in 2020. Then historical data of Digikala was applied and shown that how the model works on it.

Product customization is considered as the widespread strategy for the actual market trend oriented toward customer focus. In this field, mass customization sights mainly to emerge economy of scale and economy of scope in order to integrate mass production principles with customization abilities. This research views the collaborative management through an integrated procurement, production and distribution mixed integer linear programming (MILP) as a planning modeling approach for a multi-echelon and multi-site supply chain within tactical decision level. The model formulation is based on dyadic relationships according to leaders and followers tradeoffs where the supply chain’s stakeholders are depicted as follows, a) customers: Original Equipment Manufacturers (OEMs) identified as leaders and (b) first-tier suppliers: customized products manufacturers (c) second-tier suppliers: raw material suppliers, identified as followers. The feasibility of the proposed model has been provided through its resolution to optimality by an exact method, the decision-making process is focused on the first-tier suppliers’ operations in order to satisfy the customized demands taking into account realistic characteristics of mass customization environment for the internal and external constraints through the supply chain. The illustration of the model is performed with an example from the automotive industry, a sensitivity analysis has been conducted in order to provide the main decision points through key parameters, for instance, the capacities threshold according to a defined demand level and its customized structure which contribute to highlight a constructive managerial insights.

In this paper, we develop a multi-period mathematical model involving economic and environmental considerations. A vehicle-routing problem is considered an essential matter due to decreasing the routing cost, especially in the concerned bioenergy supply chain. A few of the optimization model recognized the vehicle routing to design the bioenergy supply chain. In this study, a bi-objective mixed-integer linear programming (MILP) model is presented.  The economic objective function minimizes the transportation, capacity expanding, fixed and variable costs, and the locating routing cost in this problem. The proposed bi-objective model is solved through a Non-Dominated Genetic Algorithm (NSGA- II).  Furthermore, the small-sized problem is solved by the CPLEX solver and augmented ɛ-constraint method.

Because of the high costs for the delivery, manufacturers are generally needed to dispatch their products in a batch delivery system. However, using such a system leads to some adverse effects, such as increasing the number of tardy jobs. The current paper investigates the two-machine flow-shop scheduling problem where jobs are processed in series on two stages and then dispatched to customers in batches. The objective is to minimize the batch delivery cost and tardiness cost related to the number of tardy jobs. First, a mixed-integer linear programming model (MILP) is proposed to explain this problem. Because the problem under consideration is NP-hard, the MILP model cannot solve large-size instances in a reasonable running time. Some metaheuristic algorithms are provided to solve the large-size instances, including BA, PSO, GA, and a novel Hybrid Bees Algorithm (HBA). Using Friedman and Wilcoxon signed-ranks tests, these intelligent algorithms are compared, and the results are analyzed. The results indicate that the HBA provides the best performance for large-size problems.

Designing a biofuel supply chain plays an important role in the reduction of biomass transportation costs. This study aims to present a comprehensive decision support tool (DST) for designing of the integrated biodiesel supply chain (BSC). In addition, so far no research has been found that examined hybrid first/second generation of biodiesel with considering all economic, environmental and social costs. In achieving this goal, we developed a new optimization model using mixed integer linear programming with the objective of maximizing the total profits of BSC incorporating environmental and social costs.  To do so, practical constraints including the limit of biomass, the capacity of technologies, the land availability, and especially limited capacity of each transportation vehicles are applied to this mathematical model. The main purpose of this study is to develop a DST to evaluate the commercial feasibility of BSC with focusing on multimodal and reliable transport.  To illustrate the capability of the proposed model, Iran is considered as a real application. The findings of this study indicate that some factors such as biomass availability, transportation reliability, and biofuel price can play as a pivotal role in this supply chain design and optimization. All in all, 31% increase in amount of produced biodiesel leads a marginal increase in environmental-related costs.

One of the main critical steps that should be taken during natural disasters is the assignment and distribution of resources among affected people. In such situations, this can save many lives. Determining the demands for critical items (i.e., the number of injured people) is very important. Accordingly, a number of casualties and injured people have to be known during a disaster. Obtaining an acceptable estimation of the number of casualties adds to the complexity of the problem. In this paper, a location-routing problem is discussed for urgent therapeutic services during disasters. The problem is formulated as a bi-objective Mixed-Integer Linear Programming (MILP) model. The objectives are to concurrently minimize the time of offering relief items to the affected people and minimize the total costs. The costs include those related to locations and transportation means (e.g., ambulances and helicopters) that are used to carry medical personnel and patients. To address the bi-objectiveness and verify the efficiency and applicability of the proposed model, the ε-constraint method is employed to solve several randomly-generated problems with CLEPX solver in GAMS. The obtained results include the objective functions, the number of the required facility, and the trade-offs between objectives. Then, the parameter of demands (i.e., number of casualties), which has the most important role, is examined using a sensitivity analysis and the managerial insights are discussed.

Multitasking is an important part of today’s manufacturing plants. Multitask machine tools are capable of processing multiple operations at the same time by applying a different set of part and tool holding devices. Mill-turns are multitask machines with the ability to perform a variety of operations with considerable accuracy and agility. One critical factor in simultaneous machining is to create a schedule for different operations to be completed in minimum make-span. A Mixed Integer Linear Programming (MILP) model is developed to address the machine scheduling problem. The adopted assumptions are more realistic when compared with the previous models. The model allows for processing multiple operations simultaneously on a single part; parts are being processed on the same setup and multiple turrets can process a single operation of a single job simultaneously performing multiple depths of cut. A Simulated Annealing algorithm with a novel initial solution and assignment approach is developed to solve large instances of the problem.

One of the problems tourism faces is how to make itineraries more effective and efficient. This research has solved the routing problem with the objective of maximizing the score and minimizing the time needed for the tourist’s itinerary. Maximizing the score means collecting a maximum of various kinds of score from each destination that is visited. The profits differ according to whether those destinations are the favorite ones for the tourists or not. Minimizing time means traveling time and visiting time in the itinerary being kept to a minimum. Those are small case with 16 tourism destinations in East Java, and large case with 56 instances consists of 100 destinations each from previous research. The existing model is the Team Orienteering Problem with Time Window (TOPTW), and the development has been conducted by adding another objective, minimum time, become Flexible TOPTW. This model guarantees that an effective itinerary with efficient timing to implement will be produced. Modification of Iterated Local Search (ILS) into Adjustment ILS (AILS) has been done by replacing random construction in the early phase with heuristic construction, continue with Permutation, Reserved and Perturbation. This metaheuristic method will address this NP-hard problem faster than the heuristic method because it has better preparation and process. Contributing to this research is a multi-objective model that combines maximum score and minimum time, and a metaheuristics method to solve the problem faster and effectively. There are calibration parameter with 17 instances of 100 destinations each, small case test using Mixed Integer Linear Programming, and large case test comparing AILS with Multi-Start Simulated Annealing (MSA), Simulated Annealing (SA), Artificial Bee Colony (ABC), and Iterated Local Search. The result shows that the proposed model will provide itinerary with less number of visited destination 4.752% but has higher total score 8.774%, and 3836.877% faster, comparing with MSA, SA, and ABC. While AILS is compared with ILS, it has less visited destination 5.656%, less total score 56.291%, and faster 375.961%. Even though AILS has more efficient running time than other methods, it needs improvement in algorithm to create better result.

The present study has developed an integrated framework for handling a facility relocation problem by combining quantitative modelling methodology with the factor rating technique. It has demonstrated the framework with reference to a real case of a corrugated box manufacturing plant. The first phase of the integrated framework involves evaluation of supply chain cost including inbound logistics cost, in-plant operations cost, and outbound logistics cost of the existing location and two other candidate locations. A mixed integer linear programming model was formulated to reflect the above problem. Sensitivity analyses were also carried out on different parameters in order to test the behaviour of the model in respect of total cost. The second phase involves evaluation of both quantitative and qualitative factors across all three locations by a team of experts on a common scale. The outcome of the first phase expressed in terms of quantitative elements of cost enabled the experts to suitably evaluate the candidate locations on cost dimension on the common scale. Finally, the composite score is computed for all three locations which aids location planners in making a realistic comparison among the three locations. Towards the end, managerial implications of the findings are discussed.

Simultaneous production planning and scheduling has been identified as one of the most important factors that affect the efficient implementation of planning and scheduling operations for the production systems. In this paper, simultaneous production planning and scheduling is applied in a hybrid flow shop environment, which has numerous applications in real industrial settings. In this problem, it is assumed that each time period includes a number of discontinuous intervals called work shifts. A novel mixed integer linear programming model is formulated. Since this problem is NP-hard in the strong sense, a new heuristic algorithm is developed to construct a complete schedule from a solution matrix that is embedded in the proposed Tabu search. A number of test problems have been solved to compare the performance of the proposed method with the exact method. The results show that the proposed tabu search is an effective and efficient method for simultaneous production planning and scheduling in hybrid flow shop systems.

The study of product family and its design as well as issues related to supply chain is as fascinating discussion, and its modeling and optimization consider as a challenge for industries and businesses. In this paper, using a consolidated approach, a comprehensive model in the Mixed Integer Linear Program (MILP) dominant is proposed to concurrent optimization of product family and its supply chain network design by considering reverse logistics. In the proposed model, different levels of bill of material, including components, sub-assemblies, sub-sub-assemblies and finished products is considered while there is possibility of substitution at all levels. The supply chain network, includes 5 levels consist of suppliers, factories, distribution centers, customers and recycling centers. To solve low complexity instances in the view of products design and supply chain network structure, CPLEX solver has been applied. To solve high complexity instances, a heuristic method based on linear programming rounding has been developed, which caused a considerable reduction in solving time with an acceptable gap.

In this article, we propose a special case of two-echelon location-routing problem (2E-LRP) in cash-in-transit (CIT) sector. To tackle this realistic problem and to make the model applicable, a rich LRP considering several existing real-life variants and characteristics named BO-2E-PCLRPSD-TW including different objective functions, multiple echelons, multiple periods, capacitated vehicles, distribution centers and automated teller machines (ATMs), different type of vehicles in each echelon, single-depot with different time windows is presented. Since, routing plans in the CIT sector ought to be safe and efficient, we consider the minimization of total transportation risk and cost simultaneously as objective functions. Then, we formulate such complex problem in mathematical mixed integer linear programming (MMILP). To validate the presented model and the formulation and to solve the problem, the latest version of ε-constraint method namely AUGMECON2 is applied. This method is especially efficient for solving multi objective integer programing (MOIP) problems and provides the exact Pareto fronts. Results substantiate the suitability of the model and the formulation.

The location-routing problem is the most significant and yet new research field in location problems that considers simultaneously vehicle routing problem features with original one for achieving high-quality integrated distribution systems in beside of the global optimum. Simultaneous pickup and delivery based on time windows are the two main characteristics of logistic management that have been used separately in most of the location routing problem in spite of their various real-life application with together. Furthermore, distribution manager always trying to create a distributed system layout along with the lowest total system cost and enhancing service levels for providing all customers satisfaction. Accordingly, in the current paper is considered the mentioned gap, that is to say the bi-objective capacitated location-routing problem based on simultaneous pickup and delivery with soft time window and multi depots (BOCLRPSPDSTW). For achieving the main goal, bi-objective mixed-integer linear programming model for BOCLRPSPDSTW, on the one hand minimizing summation of all problem costs and on the other hand, for meeting customer service level minimizing maximum summation of delivery times and service times are addressed. To solve the presented model, NSGAII and NRGA are proposed and at last efficiency of the anticipated solutions are depicted by testing them in a data set.

This research addresses the mixed-model assembly line (MMAL) by considering various constraints. In MMALs, several types of products which their similarity is so high are made on an assembly line. As a consequence, it is possible to assemble and make several types of products simultaneously without spending any additional time. The proposed multi-objective model considers the balancing and sequencing problems, simultaneously. Based on the assembly problem, the various tasks of models are assigned to the workstations, while in the sequencing problem, a sequence of models for production is determined. The two meta-heuristic algorithms, namely MOPSO and NSGA-II are used to solve the developed model and different comparison metrics are applied to compare these two proposed meta-heuristics. Several test problems based on empirical data is used to illustrate the performance of our proposed model. The results show that NSGA-II outperforms the MOPSO algorithm in most metrics used in this paper. Moreover, the results indicate that our proposed model is more effective and efficient to assignment of tasks and sequencing models than manual strategy. Finally, conclusion remarks and future research are provided.

Although several papers have studied no-idle scheduling problems, they all focus on flow shops, assuming one processor at each working stage. But, companies commonly extend to hybrid flow shops by duplicating machines in parallel in stages. This paper considers the problem of scheduling no-idle hybrid flow shops. A mixed integer linear programming model is first developed to mathematically formulate the problem. Using commercial software, the model can solve small instances to optimality. Then, two metaheuristics based on variable neighborhood search and genetic algorithms are developed to solve larger instances. Using numerical experiments, the performance of the model and algorithms are evaluated.

Over the two last decades, distribution companies have been aware of the importance of paying attention to the all aspects of a distribution system simultaneously to be successful in the global market. These aspects are the economic, the environmental, the social and the safety aspects. In the Vehicle Routing Problem (VRP) literature, the economic issue has often been used, while the environmental, the safety and the social concerns have been less proportion of studies. The Green vehicle routing problem (GVRP) is one of the recent variants of the VRP, dealing with environmental aspects of distribution systems. In this paper, two developed mixed integer programming models are presented for the GVRP with social and safety concerns. Moreover, a Genetic Algorithm (GA) is developed to deal efficiently with the problem in large size. Different numerical analyses have performed to validate the presented algorithm in comparison to exact solutions and investigate the influence of several key factors like the effect of increasing the cost of safety aspect on route balancing, and customer waiting time. The results confirm that the proposed algorithm performs well and has more social and safety benefits (such as more balanced tours and fewer customers waiting time than the classic GVRP.