Mixed-integer Linear Programming

The mixed-integer linear programming (MILP) data envelopment analysis  (DEA) models were proposed due to this reason that the conventional DEA models may suggest some non-integer values for the inputs and/or outputs of decision making units (DMUs) which those data can only be specified with integer values such as the number of employees and books. There are a few researches on integer-valued DEA and this paper focuses on the three previous ones. The paper characterizes some shortcomings on those researches with some mathematics logic.

We present a new planning model for freight consolidation carriers, one that links strategic, resource-acquisition and allocation decisions with tactical, service network design-related decisions. Specifically, like service network design models that recognize resource constraints, the model selects services and routes both commodities and the resources needed to support the services that transport them. In addition, the model recognizes that resources can be grouped into types that differ from one another with respect to capabilities, e.g. speeds, capacities, scheduling rules, etc. Ultimately, along with recognizing resource constraints, the model also makes strategic decisions such as how many resources of each type should be acquired, to what terminal new resources should be assigned, and which existing terminal-based resources should be reassigned. As such, the model can be used from a strategic planning, resource-acquisition, mixing, and allocation perspective as it provides an estimate of the impact of such decisions on transportation costs. We extend a matheuristic for a service network design problem with a fixed set (both in number and allocation) of resources of a single type to one that can also make these acquisition and allocation decisions for multiple types of resource. Then, with an extensive computational study, we demonstrate the efficacy of the matheuristic and benchmark its performance against both a leading commercial solver and a column generation-based heuristic. Finally, we perform an extensive computational study to understand how the resource-related and service network design-related components of the model interact, including how freight volumes and cost structures impact how many resources should be acquired.

This paper seeks to solve the long-term transmission expansion planning problem in power systems more effectively by reducing the solution search space and the computational effort. The proposed methodology finds and adds cutting planes based on structural insights about bus angle-differences along paths. Two lemmas and a theorem are proposed which formally establish the validity of these cutting planes onto the underlying mathematical formulations. The path-based bus angle-difference constraints, which tighten the relaxed feasible region, are used in combination with branch-and-bound to find lower bounds on the optimal investment of the transmission expansion planning problem. This work also creates an algorithm that automates the process of finding and applying the most effective valid inequalities, resulting in significantly reduced testing and computation time. The algorithm is implemented in Python, using CPLEX to add constraints and solve the exact DCOPF-based transmission expansion problem. This paper uses two different-sized systems to illustrate the effectiveness of the proposed framework: a modified IEEE 118-bus system and a modified Polish 2383-bus system.

Unit commitment (UC) is a popular problem in electric power system that aims at minimizing the total cost of power generation in a specific period, by defining an adequate scheduling of the generating units. The UC solution must respect many operational constraints. In the past half century, there was several researches treated the UC problem. Many works have proposed new formulations to the UC problem, others have offered several methodologies and techniques to solve the problem. This paper gives a literature review of UC problem, its mathematical formulation, methods for solving it and Different approaches developed for addressing renewable energy effects and uncertainties.

We investigate a difficult scheduling problem in a semiconductor manufacturing process that seeks to minimize the number of tardy jobs and makespan with sequence-dependent setup time, release time, due dates and tool constraints. We propose a mixed integer programming (MIP) formulation which treats tardy jobs as soft constraints so that our objective seeks the minimum weighted sum of makespan and heavily penalized tardy jobs. Although our polynomial-sized MIP formulation can correctly model this scheduling problem, it is so difficult that even a feasible solution can not be calculated efficiently for small-scale problems. We then propose a technique to estimate the upper bound for the number of jobs processed by a machine, and use it to effectively reduce the size of the MIP formulation. In order to handle real-world large-scale scheduling problems, we propose an efficient dispatching rule that assigns a job of the earliest due date to a machine with least recipe changeover (EDDLC) and try to re-optimize the solution by local search heuristics which involves interchange, translocation and transposition between assigned jobs. Our computational experiments indicate that EDDLC and our proposed reoptimization techniques are very efficient and effective. In particular, our method usually gives solutions very close to the exact optimum for smaller scheduling problems, and calculates good solutions for scheduling up to 200 jobs on 40 machines within 10 min.

Aggregate blending is a problem frequently encountered in the construction industry. In this article, it is shown that robust linear programming can be used to produce solutions protected against noisy data with the same computational complexity of the classic optimization
methods for the problem. It is also shown that nonlinear cost functions can be approximated by piecewise linear functions, keeping complexity at a low level. Finally, experimental results are presented. The aim is to understand how robust linear programming solutions
compare with those of other methods previously described in the literature and to evaluate the actual quality of the robust solutions.

"This paper extends the results for capacitated lot-sizing research to include pricing. Based on a few examples, the new version appears to by much easier to solve computationally. The paper, by including price, can modify demand as well as production schedule. Due to model assumptions (form of demand) a feasible solution can be found easily, unlike CLSP.
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The paper is devoted to optimal vaccination scheduling during a pandemic to minimize the probability of infection. The recent COVID-19 pandemic showed that the international community is not properly prepared to manage a crisis of this scale. Just after the vaccines had been approved by medical agencies, the policymakers needed to decide on the distribution strategy. To successfully fight the pandemic, the key is to find the equilibrium between the vaccine distribution schedule and the available supplies caused by limited production capacity. This is why society needs to be divided into stratified groups whose access to vaccines is prioritized. Herein, we present the problem of distributing protective actions (i.e., vaccines) and formulate two mixed-integer programs to solve it. The problem of distributing protective actions (PDPA) aims at finding an optimal schedule for a given set of social groups with a constant probability of infection. The problem of distributing protective actions with a herd immunity threshold (PDPAHIT) also includes a variable probability of infection, i.e., the situation when herd immunity is obtained. The results of computational experiments are reported and the potential of the models is illustrated with examples.

Unlike stick-built construction projects, structural components in prefabricated construction projects are produced in a factory environment. Thus, similar to manufacturing industries, the productivity of building components fluctuates, owing to human error and machine malfunction. Since the site demand must always be met, outsourcing manufacturing is employed to supplement the uncertain self-production. Furthermore, storage facility between factory and construction site becomes indispensable because these components are large in size, yielding a three-tier supply chain that is absent in the traditional construction. The objective of this research is to determine the most appropriate production plan and the optimal outsourcing quantities for multi-prefabricated components under this context. A workflow which includes a two-stage stochastic programming model and a mixed integer linear programming model is established for solving the above issues. Also, the most favourable schemes for distributing components and the variation of inventory are also disclosed. An infrastructure built through prefabrication method has been chosen as a case study for validation.

Over the last decades, the increasing water demand has caused a number of problems, to which reservoir operation optimization has been suggested as one of the best solutions. In this research, a model based on mixed integer linear programming (MILP) technique is developed for the systematic operation of multi-reservoirs that are used to cater for the different needs of the Tehran-Karaj plain. These reservoirs include Laar, Latian, and Karaj dams. The system configuration was accomplished through the nodes and arcs of the network flow model approach and system component implementation including sources, consumption, junctions, and the physical and hydraulic relationship between them. The following were performed via comprehensive developed software: system
configuration, objective function and constraints formulation, linearization, determining penalty values, and setting priorities for each node and arc in the system. A comparison between the MILP developed model’s results against the periodic data shows
21.7% less overflow, 11.6% more outflow, and 15.9% more reservoir storage, respectively. The outcome of theMILP-based modeling indicates superior performance to the historical period.

Business associations often promote forums between entrepreneurs from different segments. One strategy for establishing contacts among the participants is to organize them into small groups and allocate them to tables so that participants in the same group can advertise their companies, present projects and exchange ideas and experiences. After a certain time interval, a new session is started and new groups of participants are formed and allocated at other tables in order to make new contacts. The repetition of this dynamic until the closure of the forum allows each participant to make a large number of contacts. However, in order to broaden the networks of relationships it is essential to avoid re-encounters in the new groups throughout the sessions of the event. In this work we present an Integer Linear Programming model capable of organizing the sequence of tables to be visited by each participant with the objective of avoiding or minimizing re-encounters. The model was implemented in R environment and it has an interface in MS Excel for input/output data. The results generated by the model are illustrated by a computational experiment.

MAXBAND and MULTIBAND are two well-recognized methods for arterial signal control. In this paper, an Asymmetrical Multi-BAND (AM-BAND) model is developed by relaxing the symmetrical progression band requirement in MULTIBAND. Such a relaxation allows the AM-BAND model to better utilize the available green times in each direction and to provide additional opportunities for vehicular progression. Similar to MULTIBAND, the proposed AM-BAND model is formulated as a mixed-integer linear program. However, in AM-BAND the green bands in each directional section of the arterials do not have to be symmetrical with respect to the progression line. The performance of the AM-BAND model is evaluated for two arterial network data using the AIMSUN microscopic traffic simulation tool. The optimal signal coordination plans are computed by the IBM CPLEX Optimization Studio and compared with signal timing plans generated by AM-BAND, MULTIBAND, and MAXBAND. Simulation results indicate that the traffic signal coordination plans generated by AM-BAND can provide significant benefits compared to those generated by MAXBAND and MULTIBAND. Published by Elsevier Ltd. 1. Introduction Delay and number of stops are two main criteria often considered by traffic engineers for designing optimal arterial traffic signal control plans. To minimize delay and number of stops and to improve traffic operations efficiency, closely spaced traffic signals along an arterial are typically coordinated with a common cycle length and appropriate offsets such that a platoon of vehicles can travel along the entire arterial without stopping. The portion of the cycle time that allows a platoon of vehicles to traverse the arterial without stopping is termed the progression band for that direction. The width of the band is called bandwidth. By maximizing the inbound and outbound bandwidths along an arterial, vehicular progression along the arterial can be maintained and the total delay and number of stops can usually be minimized. Other than minimizing delay and number of stops, properly coordinated traffic signals can effectively increase travel speed along the arterial, mitigate traffic emissions, and reduce the probabilities for rear-end collisions. Due to its significant impact on arterial traffic control and management, traffic signal coordination has long been a popular research area in traffic engineering. Research on arterial traffic signal coordination can generally be divided into two categories: one is maximizing the bandwidth along an arterial; the other is minimizing the total delay for the signalized system. Bandwidth-based models perform best when the arterial through traffic volume is dominant compared with the turning volume and the cross street